arXiv:1511.06913v4 [] 26 Jun 2017

Hyperbolic actions and 2nd bounded cohomology
of subgroups of Out(Fn)
Part I: Infinite lamination subgroups
Michael Handel and Lee Mosher

∗

June 27, 2017

Abstract
In this two part work with sequel [HM17a] we prove that for every finitely
generated subgroup Γ < Out(Fn ), either Γ is virtually abelian or Hb2 (Γ; R)
contains an embedding of ℓ1 . The method uses actions on hyperbolic spaces,
for purposes of constructing quasimorphisms. Here in Part I, after presenting
the general theory, we focus on the case of infinite lamination subgroups Γ
— those for which the set of all attracting laminations of all elements of Γ is
infinite — using actions on free splitting complexes of free groups.

1

Introduction

The study of hyperbolic actions — group actions on Gromov hyperbolic spaces —
has co-evolved with the study of the 2nd bounded cohomology Hb2 (Γ; R) of a group Γ.
This started with Brooks’ theorem, using the action of a free group Γ of rank ≥ 2 on its
Cayley tree to prove that there is an embedding ℓ1 ֒→ Hb2 (Γ; R) [Bro81]. In works to
follow, the same conclusion for Hb2 (Γ; R) was proved in increasing generality for certain
hyperbolic actions that are proper(ly discontinuous) [BS84], [EF97], [Fuj98]. Fujiwara
extended the method to work for certain nonproper actions [Fuj00]. Bestvina and
Fujiwara [BF02], using hyperbolicity of the curve complex C(S) of a finite type surface
S [MM99], proved an “Hb2 -alternative” for subgroups Γ < MCG(S) of the mapping
class group: either Γ is virtually abelian, or there is an embedding ℓ1 ֒→ Hb2 (Γ; R).
While the action on C(S) is not proper, nonetheless Bestvina and Fujiwara distilled
enough proper discontinuity to generalize Hb2-methods, by introducing the WPD or
“weak proper discontinuity” property; and a really weak but still useful version of
The first author was supported by the National Science Foundation under Grant No. DMS1308710 and by PSC-CUNY under grants in Program Years 46 and 47. The second author was
supported by the National Science Foundation under Grant No. DMS-1406376.
∗

1

WPD known as WWPD was introduced by Bestvina, Bromberg, and Fujiwara in
[BBF15], for their study of the asymptotic dimension of MCG(S). For later steps of
this co-evolution, using WPD to further the study of hyperbolic actions, see [Bow08],
[Osi16], and [BHS14].
For subgroups Γ < Out(Fn ), Bestvina and Feighn [BF10] produced enough WPD
hyperbolic actions to prove the Hb2-alternative if Γ contains a fully irreducible outer
automorphism; for another proof of this result by Hamenstädt see [Ham14]. Applying
our early subgroup decomposition theory [HM09], the same conclusion follows if Γ
is finitely generated and fully irreducible (as defined in Section 4.1). By work of
Horbez [Hor16] using different methods, the conclusion follows for all fully irreducible
subgroups, without assuming finite generation.
Here is our main result, to be proved over Parts I and II of this work:
Theorem A. Finitely generated subgroups of Out(Fn ) satisfy the Hb2 -alternative.
Theorem A is proved using Theorems B and C regarding constructions of hyperbolic actions and WWPD elements, and Theorem D regarding WWPD methods that
generalize WPD methods of [BF02]. To state these results, we need some definitions.
WWPD. Given an action G y X on a hyperbolic space, an element γ ∈ G
is loxodromic if its action on the Gromov boundary ∂X has north-south dynamics,
with repeller–attractor pair ∂± γ = (∂− γ, ∂+ γ) ∈ ∂X × ∂X − ∆. The action G y X
is nonelementary if there exists a pair δ, γ ∈ G of loxodromic elements which are
independent meaning that {∂− δ, ∂+ δ} ∩ {∂− γ, ∂+ γ} = ∅. An element γ ∈ G satisfies
WWPD if γ is loxodromic and, under the natural induced action G y ∂X × ∂X − ∆,
the orbit G · ∂± γ is discrete in ∂X ×∂X −∆. The WPD condition requires in addition
that the subgroup of G that stabilizes ∂± γ be virtually cyclic.
Finite and infinite lamination subgroups. Behind the results of [BF02] are
Thurston’s decomposition theory for elements of MCG(S) [FLP+ 12], Ivanov’s decomposition theory for subgroups of MCG(S) [Iva92], and the Masur-Minsky results on
hyperbolicity of C(S) [MM99]. Given φ ∈ MCG(S), its lamination set L(φ) is a finite set consisting of one unstable lamination for each pseudo-Anosov piece of the
Thurston decomposition of φ. Each subgroup Γ < MCG(S) has its associated lamination set L(Γ) = ∪φ∈Γ L(φ) which is useful in applying subgroup decomposition theory.
For example, Γ < MCG(S) is virtually abelian if and only if L(Γ) is finite.
Behind the proof of Theorem A are the decomposition theory for elements of
Out(Fn ) due to Bestvina, Feighn, and Handel [BFH97, BFH00, BFH05, FH11], a
decomposition theory for abelian subgroups [FH09] due to Feighn and Handel, our
decomposition theory for general finitely generated subgroups, [HM17b]–[HM17f],
and our results on hyperbolicity and dynamics of the free splitting complex F S(Fn )
[HM13, HM14b]. Associated to each φ ∈ Out(Fn ) is its finite set L(φ) of attracting
laminations [BFH00]. Each subgroup Γ < Out(Fn ) has its associated lamination set
L(Γ) = ∪φ∈Γ L(φ). If L(Γ) is finite then Γ is a finite lamination subgroup, otherwise
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it is an infinite lamination subgroup. Every virtually abelian subgroup of Out(Fn ) is
a finite lamination subgroup, but the converse does not hold, unlike in MCG(S).
Subgroups of IAn (Z/3) with (virtually) abelian restrictions. In [BF02],
Bestvina and Fujiwara use Ivanov’s results [Iva92] to reduce to the case of an “irreducible” subgroup Γ < MCG(S) — one having no invariant curve system. We also
reduce to special subgroups of Out(Fn ) as stated in the heading just above, a reduction
of a somewhat different nature. Recall the finite index, characteristic subgroup

IAn (Z/3) = IA(Fn ; Z/3) = Ker Out(Fn ) 7→ GL(n, Z/3)

In Section 4.1 we review the following important invariance properties of IAn (Z/3).
First, IAn (Z/3) is torsion free ([BT68] and see [Vog02]). Next, if φ ∈ IAn (Z/3) then
the conjugacy class of any element or free factor of Fn which is φ-periodic is fixed by φ
[HM17d]. Finally, every virtually abelian subgroup of IAn (Z/3) is abelian [HM17g],
which we sometimes emphasize by writing “(virtually) abelian” in the context of a
subgroup of IAn (Z/3). In various theorems we often restrict to subgroups of IAn (Z/3)
in order to have these properties, and in applying such theorems we achieve this
restriction by replacing a subgroup by its intersection with IAn (Z/3).
For any free factor B < Fn with conjugacy class [B] and stabilizer subgroup
Stab[B] < Out(Fn ), there is a natural restriction homomorphism Stab[B] 7→ Out(B)
[HM17c, Fact 1.4]. A subgroup Γ < IAn (Z/3) has (virtually) abelian restrictions
if for any proper free factor B < Fn such that Γ < Stab[B], the restriction map
Γ 7→ Out(B) has (virtually) abelian image. The two versions of this property — with
or without “virtually” — are equivalent by [HM17g] (see Corollary 4.1 for details).
Among finitely generated subgroups of IAn (Z/3) with (virtually) abelian restrictions, Theorems B and C are concerned, respectively, with infinite lamination and
finite lamination subgroups, and their actions on hyperbolic spaces. Theorem B uses
the free splitting complex F S(Fn ), hyperbolicity of which is proved in [HM13], and
loxodromic elements of which are characterized in [HM14b]; in particular, every element of Out(Fn ) acts elliptically or loxodromically on F S(Fn ).
Theorem B. For any infinite lamination subgroup Γ < IAn (Z/3) with (virtually)
abelian restrictions, and any maximal, Γ-invariant, proper free factor system A, we
have:
(1) The action Γ y F S(Fn ) is nonelementary.
(2) There exists a loxodromic φ ∈ [Γ, Γ] which is fully irreducible rel A, and any
such φ is a WWPD element for the action of Γ on F S(Fn ).
Theorem C. For any finitely generated, finite lamination subgroup Γ < IAn (Z/3)
which is not (virtually) abelian and has (virtually) abelian restrictions, there is a finite
index normal subgroup N < Γ and there is an action N y X on a hyperbolic space,
such that the following hold:
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(1) Every element of N acts elliptically or loxodromically on X.
(2) The action N y X is nonelementary.
(3) Every element of [N, N] is either elliptic or WWPD with respect to the action
N y X.
The proof of Theorem C, in which some new hyperbolic actions are constructed, is
found in Part II. Here in Part I, Theorem C is used as a black box for purposes of
proving Theorem A.

WWPD methods and Theorem D.
Reducing Theorem A to Theorems B and C is carried out in Section 4.3, using
WWPD methods that are summarized in Theorem D. Here is a brief description of
those methods (see also the Remark at the end of Section 2.4, regarding how to avoid
Theorem D at the expense of weaker conclusions).
Given a nonelementary action of a group Γ on a hyperbolic space, the WPD property comes in an individual version defined for one loxodromic element at a time, and
in a global version which simply asserts that all loxodromic elements satisfy the individual version (see Section 2.1). In the special setting of the action MCG(S) y C(S),
that “local–global” distinction was unimportant because every loxodromic element of
the action MCG(S) y C(S) — every pseudo-Anosov mapping class — satisfies WPD.
For a general hyperbolic action action, the individual and global versions of WPD are
connected by a theorem of Osin [Osi16] saying that if a given nonelementary hyperbolic action of Γ has a WPD element then Γ has a nonelementary hyperbolic action
on a possibly different space that satisfies the global WPD property (even better,
Γ has an acylindrical hyperbolic action).
Although WPD suffices for studying Hb2 properties of subgroups of mapping class
groups [BF02], and although it suffices for subgroups of Out(Fn ) containing a fully
irreducible element [BF10], we do not know whether it suffices for general subgroups of
Out(Fn ). Instead our proof of Theorem B uses the loxodromic elements of the action
Out(Fn ) y F S(Fn ); these form a strictly larger collection than the fully irreducible
outer automorphisms, and some of them fail to satisfy WPD [HM14b]. What our
proof shows is that sufficiently many of these loxodromics satisfy the weaker WWPD
property so that Hb2 methods can be applied.
We study WWPD first as an individual property (Section 2.2), and then as a global
property (Section 2.5). The global WWPD property is designed to weaken several
different aspects of the global WPD property, retaining enough properties so that the
quasimorphism constructions found in [BF02] will still work. First, we do not require
a hyperbolic action of Γ itself, only a hyperbolic action N y X of a finite index
normal subgroup N ⊳ Γ. Second, we do not require every loxodromic of N to satisfy
WPD or WWPD, only that WWPD hold for all the nontrivial elements of some
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Schottky subgroup F ⊂ [N, N] of the commutator subgroup. The global WWPD
property also imposes similar conditions on the actions of N that are obtained by
pre-composing the given action of N with inner automorphisms of Γ. For full details
see Definition 2.13.
Theorem D. If a group Γ satisfies the global WWPD property then Hb2 (Γ; R) contains
an embedded ℓ1 .
The proof of Theorem D (see the appendix) follows closely the methods of [Fuj00]
and [BF02], combined with an application of Kalujnin-Krasner embeddings into
wreath products in order to avoid any special wreath product hypothesis as was
needed in [BF02].

Methods of proof of Theorem B.
Consider a subgroup Γ < IAn (Z/3) satisfying the hypotheses of Theorem B, so Γ
is an infinite lamination subgroup with (virtually) abelian restrictions. Consider
also a maximal, proper, Γ-invariant free factor system A. In Section 4.2, we attack
Theorem B using lamination ping-pong methods from [HM17f] to produce φ ∈ Γ
which is irreducible relative to A, has a filling attracting lamination, and is in the
commutator subgroup of Γ. The element φ acts loxodromically on F S(Fn ) because,
as proved in [HM14b], the elements of Out(Fn ) acting loxodromically on F S(Fn ) are
precisely those which have a filling attracting lamination (a strictly weaker property
than being fully irreducible, which is the criterion for loxodromic action on the free
factor complex [BF14]). Since the image of the restriction homomorphism Γ 7→
Out(A) is abelian, and since φ is in the commutator subgroup of Γ, the restriction
φ Out(A) is trivial.
Theorem B is thus reduced to the following theorem, the reduction argument
being carried out in full detail in Section 4.2.
Theorem E. Let n ≥ 3, let Γ be a subgroup of IAn (Z/3) that preserves a (possibly
empty) proper free factor system F , and let φ ∈ Γ have the following properties:
Relative irreducibility: φ is irreducible rel F ;
Trivial restrictions: φ Out(A) is trivial for each component [A] ∈ F ;
Filling lamination: φ has a filling attracting lamination, equivalently φ acts loxodromically on F S(Fn ) [HM14b].
Then φ is a WWPD element for the action of Γ on F S(Fn ).
The proof of Theorem E itself is carried out in Section 6. That proof depends on
results about well functions in the free splitting complex, developed in [HM14b] in
analogy to the well functions of Algom-Kfir in the context of outer space [AK11], and
further developed in Section 5. In particular, Section 5.4 describes regularity properties of attracting laminations which we use in place of measure theoretic properties
5

of currents as applied in [BF02] and [BF10]. Underlying Section 5.4 is a study of
uniform splitting properties of EG strata of relative train track maps carried out in
Section 3.2, strengthening splitting properties derived in earlier works.
In Theorems B and E, the free factor system F is empty if and only if the subgroup H is fully irreducible, in which case we recover the result of Bestvina and Feighn
[BF10] with a different proof. There might be various ways to relax the assumption
that F be empty. The way expressed in Theorem E—allowing F to be nonempty but
requiring H to be abelian when restricted to each component of F —was chosen because it is sufficient for purposes of proving Theorem B, and because the laminations
involved in Theorem E have a particularly simple structure that is easier than general laminations. For a description of that structure see the heading The topological
structure of Λ−
φ , under Case 2b of Section 6.3.

Application to the Bridson-Wade Theorem
As a corollary to Theorem A we prove the following theorem of Bridson-Wade, following the same lines as the analogous proof for mapping class groups found in [BF02].
Corollary ([BW11]). If Γ is an irreducible lattice in a connected, semisimple Lie
group of real rank ≥ 2 having finite center, then every homomorphism h : Γ → Out(Fn )
has finite image.
Proof. By the Margulis-Kahzdan normal subgroup theorem [Zim84, Theorem 8.1.2],
K = Ker(h) is either finite or of finite index in Γ. Assuming K is finite we derive a
contradiction.
If Γ is a nonuniform lattice then it contains a solvable subgroup H which does not
surject with finite kernel onto an abelian group, so the group H/H ∩ K is solvable
and not virtually abelian, but it injects to Out(Fn ) contradicting [BFH04, Ali02].
If Γ is a uniform lattice then it has a free subgroup of rank ≥ 2 [Mar91, Theorem
5.6], so the group Γ/K is not virtually abelian. Also, Γ is finitely generated. But Γ/K
injects to Out(Fn ) and so, by Theorem A, Hb2 (Γ/K; R) contains an embedded ℓ1 . In
particular there is an unbounded quasimorphism Γ/K 7→ R (see Section 2.1), and by
composition we get an unbounded quasimorphism Γ 7→ Γ/K 7→ R, contradicting the
theorem of Burger and Monod [BM99, Corollary 1.3].

Outline of the logic
To summarize, we shall reduce Theorem A to Theorem C by the following outline:
• Section 4.2: Proof that Theorem E implies Theorem B.
• Section 4.3: Proof that Theorems B, C, D imply Theorem A.
• Section 6: Proof of Theorem E.
• Section 7 (Appendix): Proof of Theorem D.
In Part II [HM17a] will be found the proof of Theorem C.
6

Contents
1 Introduction

1

2 Weakening weak proper discontinuity
2.1 Basic concepts. . . . . . . . . . . . . . . . . . .
2.2 WPD and WWPD . . . . . . . . . . . . . . . .
2.3 Equivalence g ∼ h in a hyperbolic group action
2.4 Individualized WWPD methods. . . . . . . . . .
2.5 Global WWPD methods: Statement of Theorem

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3 Background material on Out(Fn )
3.1 Basic concepts of Out(Fn ) . . . . . . . . . . . . . . . . . . . . . . . .
3.2 A uniform splitting lemma . . . . . . . . . . . . . . . . . . . . . . . .

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4 Reducing
4.1 Weak
4.2 Proof
4.3 Proof

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Theorem A to Theorems C, D, E
attraction and subgroup decomposition theory . . . . . . . . .
that Theorem E implies Theorem B. . . . . . . . . . . . . . . .
that Theorems B, C, D imply Theorem A . . . . . . . . . . . .

5 Well functions and weak tiling functions
5.1 Well functions on intermediate conjugacy classes . . . .
5.2 Weak tiling functions on intermediate conjugacy classes
5.3 Coarse equivalence of well functions and
weak tiling functions. . . . . . . . . . . . . . . . . . . .
5.4 Regularity properties of weak tilings. . . . . . . . . . .

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6 Proof of Theorem E
6.1 Free splitting complex and spine of relative outer space. . . . . . . . .
6.2 Setting up the proof of Theorem E. . . . . . . . . . . . . . . . . . . .
6.3 The case analysis of Theorem E. . . . . . . . . . . . . . . . . . . . . .

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7 Appendix: Proof of Theorem D

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2

Weakening weak proper discontinuity

This section contains a review of the WPD and WWPD properties, together with a
further development of WWPD culminating with the definition of the global WWPD
property and the statement of Theorem D, the proof of which is found in Section 7 (Appendix).
Section 2.1 contains a review of certain basic notions needed throughout the paper.
Section 2.2 reviews the WPD property of [BF02] and the WWPD property of [BBF15];
Proposition 2.6 establishes several equivalent formulations of WWPD, and proves
various properties of WWPD. In Section 2.3 we review the equivalence relation g ∼ h
on loxodromic elements of a hyperbolic action, which was crystallized by Bestvina
and Fujiwara [BF02], and which plays a central role in proofs that second bounded
cohomology contains an embedded ℓ1 ; Lemma 2.10 describes how WWPD and the
equivalence relation g ∼ h interact with each other. The emphasis in Section 2.3 is
to interpret WPD, WWPD, and the relation g ∼ h in terms of induced actions on
the Gromov boundary of a hyperbolic space.
The rest of this section further develops the theory of WWPD. In Section 2.4 we
will explain how to use the existence of an individual WWPD element of a nonelementary hyperbolic group action Γ y X to deduce that Hb2 (Γ; R) contains an embedded ℓ1 . In Section 2.5 we formulate a global WWPD property of a hyperbolic group
action, we study how this property reduces under passage to finite index subgroups
and quotient groups, and we use this property to state Theorem D, whose proof will
be found in the appendix.

2.1

Basic concepts.

Metrics. A metric complex is a connected simplicial complex equipped with the
geodesic metric making each k-simplex isometric to a regular k-simplex in Rk+1 of
edge lengths equal to 1, using barycentric coordinates to define the isometry. A
hyperbolic complex X is a Gromov hyperbolic metric complex; equivalently, the 1skeleton is Gromov hyperbolic. The Gromov boundary is denoted ∂X, and we denote
X = X ∪ ∂X with its Gromov topology. The space of two point subsets is denoted
∂ 2 X = {{ξ, η} ξ 6= η ∈ ∂X} with topology induced by the 2-1 covering map
(∂X × ∂X) − ∆ 7→ ∂ 2 X, equivalently the Hausdorff topology on compact subsets
of ∂X.
Remark. Using the Rips complex one easily sees that every geodesic metric space
is quasi-isometric to some metric complex [Gro87]. Since the spaces we use in this
work are all metric complexes, we couch our results in that language.
Actions. An action of a group Γ on an object X is given by a homomorphism
A : Γ 7→ Isom(X) from Γ to the self-isomorphism group of X (we use this definition
primarily for metric complexes and occasionally for objects in other categories). We
8

often use “action notation” A : Γ y X, usually suppressing A and writing simply
Γ y X. For g ∈ Γ and a subset Y ∈ X, we use expressions like g(Y ) or g · Y
for A(g)(Y ). Given an action Γ y X, the stabilizer of a subset Y ⊂ X is denoted
Stab(Y ) = Stab(Y ; Γ) = {g ∈ Γ g · Y = Y }.
Consider a hyperbolic complex X. Any isometric action Γ y X extends to a
topological action Γ y X = X ∪ ∂X. An isometry h : X → X is elliptic if each orbit
{hi · x} is of bounded diameter in X, it is loxodromic if each orbit map i 7→ hi · x
is a quasi-isometric embedding Z 7→ X, and otherwise it is parabolic. For h to be
loxodromic is equivalent to saying that the extension h : X → X has source-sink or
north-south dynamics, meaning that there is a repelling fixed point ∂− h ∈ ∂S, an
attracting fixed point ∂+ h ∈ ∂S, and for each x ∈ S − {∂− h, ∂+ h} the sequence
hi (x) converges to ∂− h as i → −∞ and to ∂+ h as i → +∞. As ordered and as
unordered pairs we denote these fixed points as ∂± h = (∂− h, ∂+ h) ∈ ∂S × ∂S − ∆ and
∂h = {∂− h, ∂+ h} ∈ ∂ 2 S. Note also that h is loxodromic if and only if the action of h
on S has a quasi-axis which is a quasi-isometric embedding γ : R → S such that for
some T > 0 called the period we have γ(t + T ) = h(γ(t)), t ∈ R; we can always take γ
to be a bi-infinite edge path in the 1-skeleton of X. A quasi-axis is determined by its
restriction γ [0, T ], subject to the requirement that h(γ(0)) = γ(T ); that restriction,
and any of its translates γ [(k − 1)T, kT ], are called fundamental domains for the
action of h on its quasi-axis.
Two loxodromic isometries g, h : X → X are independent if ∂h∩∂g = ∅. An action
Γ y X on a hyperbolic complex is elementary if it does not contain an independent
pair of loxodromic elements, and it is elliptic if every element is elliptic.
Bounded cohomology. As in other works cited earlier, we study second bounded
cohomology of a group via quasimorphisms on the group; see for example the introduction to [BF02]. We recall the precise connection between these two concepts,
which depends on basic definitions of cohomology theory and the snake lemma.
Consider a group G and its cochain complex C ∗ (G; R), where C k (G; R) is the
vector space of functions f : Gk 7→ R, with the standard coboundary operator
δ : C k (G; R) → C k+1 (G; R) [Bro82]. We need only the formula for δ : C 1 (G; R) →
C 2 (G; R) which is δ(f )(g1 , g2 ) = f (g1 ) + f (g2 ) − f (g1 g2 ). The bounded cohomology
Hbk (G; R) is the cohomology of the subcomplex of bounded cochains Cbk (G; R) = {f ∈
C k (G, R) f is bounded}.
The vector space of quasimorphisms of G is defined to be
QH(G; R) = {f ∈ C 1 (G; R) δf ∈ Cb2 (R)}
The defect of f ∈ QH(G; R) is the real number sup |δf (g1 , g2 )|.
g
The quotient space QH(G,
R) is defined by modding out QH(G; R) by the subspace spanned by those f : G 7→ R which are either actual homomorphisms or
9

bounded functions. One obtains an exact sequence
g
0 → QH(G;
R) →Hb2 (G; R) → H 2 (G; R)

by applying the snake lemma to the short exact sequence of cochain complexes of the
form
1 7→Cb∗ (G; R) → C ∗ (G; R) → C ∗ (G; R)/Cb∗(G; R) → 1
Thus to prove that Hb2 (G; R) contains an embedded ℓ1 , it is sufficient to prove the
g
same for QH(G;
R), and that is the method we will use.

2.2

WPD and WWPD

In this section we fix a group action Γ y X on a hyperbolic complex with associated
Gromov extension Γ y X = X ∪ ∂X.
Definition 2.1. Given h ∈ Γ we say that h is individually WPD with respect to Γ
if h is loxodromic and for every x ∈ X and R > 0 there exists an integer M > 0 such
that any subset Z ⊂ Γ satisfying the following property is finite:
• For each g ∈ Z we have d(x, g(x)) < R and d(hM (x), ghM (x)) < R.
Definition 2.2 ([BF02] Section 3). The action Γ y X is said to satisfy WPD if Γ is
not virtually cyclic, Γ contains at least one loxodromic element, and every loxodromic
element of Γ is individually WPD with respect to Γ.
In these definitions we usually drop “with respect to Γ” when it is clear from context.
It is shown in the proof of [BF02] Proposition 6 that if h ∈ Γ satisfies the individual
WPD property then its endpoint pair ∂± h ∈ ∂X ×∂X −∆ has discrete orbit under the
diagonal action of Γ on ∂X ×∂X −∆ and the stabilizer subgroup of the endpoint pair
∂± h is virtually infinite cyclic. The converse also holds, as is well known to experts,
and we make this explicit in Corollary 2.7 below. For purposes of formulating the
WWPD property we focus on discreteness:
Definition 2.3 (Equivariantly discrete fixed set). We say that a loxodromic element
h ∈ Γ has equivariantly discrete fixed set if the Γ-orbit of ∂h in the space ∂ 2 X is a
discrete subset, equivalently the Γ-orbit of ∂± h in the space ∂X ×∂X −∆ is a discrete
subset.
The WWPD property, formulated by Bestvina, Bromberg, and Fujiwara [BBF15],
has several equivalent reformulations given in Proposition 2.6 below. One may think
of WWPD as a fragment of the individual WPD property that is designed to allow
for arbitrary behavior of the subgroup Stab(∂± h), while still capturing the essential
discreteness concepts of WPD that are useful for applications to second bounded
cohomology and other purposes. The idea is that instead of counting the total number
of group elements that move point pairs a small amount, one counts only the number
of left cosets of Stab(∂± h) represented by such group elements.
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Definition 2.4. Given a group action Γ y X on a hyperbolic space, an element
h ∈ Γ satisfies WWPD with respect to Γ if h is loxodromic and for every x ∈ X and
R > 0 there exists an integer M ≥ 1 such that any subset Z ⊂ Γ that satisfies the
following two properties is finite:
(1) For each g ∈ Z we have d(x, g(x)) < R and d(hM (x), ghM (x)) < R; and
(2) No two elements of Z lie in the same left coset of Stab(∂± h).
As with Definition 2.1 and 2.2, we usually drop “with respect to Γ” if it is clear from
context.
Combining Definitions 2.2 and 2.4 we immediately have:
Corollary 2.5. If a loxodromic h ∈ Γ satisfies WPD then it satisfies WWPD.
In the following proposition, item (2) clarifies that when generalizing from WPD
to WWPD, what one loses control over is the structure of the group Stab(∂± h); what
one retains is precisely the discrete topology of the orbit of ∂± h. Also, item (3) is
how WWPD was originally formulated in [BBF15]. Item (4) is what we will actually
apply in the proof of Theorem E; see the heading “Setting up the proof of Theorem
E” in Section 6.
Proposition 2.6. Given a loxodromic element h ∈ Γ, the following are equivalent:
(1) h satisfies WWPD.
(2) h has equivariantly discrete fixed set.
(3) For any quasi-axis ℓ of h there exists D ≥ 0 such that for any g ∈ Γ, if g 6∈
Stab(∂± h) then the image of a closest point map g(ℓ) 7→ ℓ has diameter ≤ D.
(4) In the group Γ there is NO infinite sequence g1 , g2 , g3 , . . . satisfying the following
properties:
(a) For all i 6= j the elements gi , gj lie in different left cosets of Stab(∂± h).
(b) For all (there exists) x ∈ X there exists R > 0 such that for all M ≥ 0 there
exists I ≥ 0 such that if 0 ≤ m ≤ M and if i ≥ I then d(gi hm (x), hm (x)) <
R.
Furthermore if h satisfies WWPD then:
(5) Stab(∂− h) = Stab(∂+ h) = Stab(∂± h);
(6) If k ∈ Γ − Stab(∂h) then h and khk −1 are independent.

11

Remark: The two versions of item (4b) — one with quantifier “for all” and the other
with “there exists” — are equivalent, as we shall show in the course of the proof.
Before turning to the proof of Proposition 2.6, we first apply it to prove two
corollaries:
Corollary 2.7. Given a loxodromic h ∈ Γ, the following are equivalent:
(1) h is individually WPD.
(2) The subgroup Stab(∂± h) < Γ is virtually cyclic, and h has equivariantly discrete
fixed set.
Proof. Assuming (1) holds, the proof that Stab(∂± h) < Γ is virtually cyclic can be
found in the proof of [BF02] Proposition 6, which uses only the individual WPD
property for h. To verify that h has equivariantly discrete fixed set one can combine
Corollary 2.5 and Proposition 2.6. Thus (2) is proved.
Assuming (2) holds, it follows that h satisfies WWPD (by Proposition 2.6 (2) =⇒
(1)). Using that the subgroup Stab(∂± h) is virtually cyclic, its cyclic subgroup hhi
has finite index, and since h is loxodromic the action hhi y S is metrically proper it
follows that the action Stab(∂± h) y S is metrically proper. Fixing x ∈ S and R > 0,
consider for each integer N > 0 the following set (as in Definitions 2.2 and 2.4):
ZN = {g ∈ Γ d(x, g(x)) < R and d(hN (x), ghN (x)) < R}
Since h satisfies WWPD, there exists N such that ZN intersects only finitely many
left cosets of the subgroup Stab(∂± h). But since the action of Stab(∂± h) is metrically
proper, each of its left cosets has finite intersection with ZN . The set ZN is therefore
finite, showing that h satisfies WPD and so (1) is proved.
The next result generalizes Corollary 2.5:
Corollary 2.8. Consider a group action Γ y X on a hyperbolic space with image
subgroup Q = Image(Γ 7→ Isom(X)), an element h ∈ Γ, and the image q ∈ Q of
h under the quotient homomorphism Γ 7→ Q. If q is a WPD element of the action
Q y X then h is a WWPD element of the action Γ y X. More generally (by
Corollary 2.5), if q is a WWPD element of Q y X then h is a WWPD element of
Γ y X.
Proof. Note that h is loxodromic, that ∂± h = ∂± q, and that the orbits H · ∂± h
and Q · ∂± q are the same subset of ∂X × ∂X − ∆. Applying the equivalence of
items (1), (2) in Proposition 2.6, and using that q is WWPD, it follows first that
the set H · ∂± h = Q · ∂± q is discrete in ∂X × ∂X − ∆, and then that h satisfies
WWPD.
We finish this section with:
12

Proof of Proposition 2.6. In this proof we adapt interval notation [x, y] to denote a
choice of geodesic between points x, y ∈ X in a geodesic metric space. Usually we
use function notation for quasigeodesics, such as γ : [s, t] → X, but we also briefly
use the notation xy where x = γ(s), y = γ(t).
The equivalence (2) ⇐⇒ (3) is elementary; here is a sketch with vague constants.
Let ℓ be a quasi-axis for h. If (2) fails then there is a sequence gi ∈ Γ−Stab(∂± h) such
that gi (∂± h) = gi (∂± ℓ) converges to ∂± h = ∂± ℓ. It follows that there are sequences
xi , yi ∈ ℓ such that xi → ∂− h, yi → ∂+ h, and xi , yi are within uniformly bounded
distance of gi (ℓ). The image of the closest point projection map πi : gi (ℓ) 7→ ℓ therefore
has points within uniformly bounded distance of xi , yi , and since d(xi , yi ) → +∞ it
follows that diam(Image(πi )) → +∞, and so (3) fails. Conversely if (3) fails then
there is a sequence gi ∈ Γ − Stab(∂± h) such that, letting πi : gi (ℓ) → ℓ be a closest
point map, the diameter of Image(πi ) goes to +∞ with i. After postcomposing each
gi with an appropriate power of h, Image(πi ) comes uniformly Hausdorff close to an
entire subsegment [xi , yi ] of ℓ such that xi → ∂− h and yi → ∂+ h. Furthermore, there
is a subsegment [x′i , yi′ ] of gi (ℓ) which is uniformly Hausdorff close to [xi , yi ]. It follows
that ∂− gi (ℓ) → ∂− h and ∂+ gi (ℓ) → ∂+ h, proving that (2) fails.
Item (4) asserts that there does not exist an infinite sequence satisfying the conjunction of (4a) and (4b). Let (4)∀ denote item (4) with the quantifier at the beginning
of (4b) specified to be “for all”, and similarly for (4)∃ . Since (4) is a negation, clearly
(4)∃ =⇒ (4)∀ . It remains to prove the chain of implications (4)∀ =⇒ (1) =⇒ (2)
=⇒ (4)∃ .
For the rest of the proof, let L(k, c, δ) ≥ 0 be a thinness constant for k, c quasigeodesic quadrilaterals in a δ-hyperbolic space X, with finite or infinite endpoints:
for any w, x, y, z ∈ X and any k, c quasigeodesics wx, xy, yz, zw, the quasigeodesic
wx is contained in the L-neighborhood of xy ∪ yz ∪ zw. Note for example that
L(1, 0, δ) = 2δ.
We turn to the proof that (4)∀ =⇒ (1). Fix δ to be a hyperbolicity constant
for X. Suppose that h fails to satisfy (1), i.e. h is not WWPD, so there exists x ∈ X
and R0 > 0, and there exist sequences Mi → +∞ in the positive integers and gi in Γ,
such that the elements gi all lie in different left cosets of Stab(∂± h) (thus satisfying
item (4a)), and such that d(x, gi (x)) < R0 and d(hMi (x), gi hMi (x)) < R0 . Since h is
loxodromic, the sequence m 7→ hm (x) is a quasi-isometric embedding Z 7→ X and can
be interpolated to obtain a k, c quasi-axis for h of the form
· · · [h−2 (x), h−1 (x)] ∗ [h−1 (x), h0 (x)] ∗ [h0 (x), h1 (x)] ∗ [h1 (x), h2 (x)] ∗ · · ·
Since gi moves the points x = h0 (x) and hMi (x) a distance at most R0 , by an elementary argument in Gromov hyperbolic geometry there exists a constant R depending
only on R0 , k, c, and δ such that all points on this segment between x = h0 (x) and
hMi (x) are moved by gi a distance at most R (independent of i). Thus item (4b)∃ is
satisfied, and so h fails to satisfy (4)∀ .
13

We next prove that (2) =⇒ (4)∃ . Assuming that (4)∃ fails, we may pick an infinite
sequence of elements f1 , f2 , f3 , . . . lying in different left cosets of Stab(∂± h), and we
may pick x ∈ X and R > 0, so that the following holds: for all M ≥ 0 there exists
I ≥ 0 such that if 0 ≤ m ≤ M and if i ≥ I then d(fi hm (x), hm (x)) < R. It follows
that there exists a growing sequence of positive even integers mi = 2ki → +∞ such
that for each i the isometry fi moves each of the points x, h(x), h2 (x) . . . , h2ki (x)
a distance at most R. Let gi = h−ki fi hki , and note that gi moves each of the
points h−ki (x), h−ki +1 (x), . . . , hki −1 (x), hki (x) a distance at most R. Since the sequence k 7→ hk (x) defines a quasi-isometric embedding Z 7→ X, it follows that the
sequence (h−ki (x), hki (x)) converges to ∂± h in the space of ordered pairs of points
in X ∪ ∂X, and therefore the sequence (g(h−ki (x)), g(hki (x))) converges to ∂± h, and
therefore the sequence ∂± (gi hgi−1 ) = gi (∂± h) converges to ∂± h. We may assume,
dropping at most a single term of the sequence, that fi 6∈ Stab(∂± h) for each i, and
hence gi 6∈ Stab(∂± h). Since gi (∂± h) 6= ∂± h but gi (∂± h) converges to ∂± h, we have
proved that h does not have equivariantly discrete fixed set and so (2) fails.
It remains to prove that (1) =⇒ (2). Assuming that h does not have equivariantly
discrete fixed set, there is a sequence fi ∈ Γ such that fi (∂± h) 6= ∂± h and such that
fi (∂± h) → ∂± h as i → +∞. Notice that for some values of i we might have fi (∂− h) =
∂− h and for others fi (∂+ h) = ∂+ h, although those cannot happen simultaneously. Let
γ : R → X be a k, c quasi-axis for h with period T , let L = L(k, c, δ), and let ∆ be
diameter of the fundamental domain γ[0, T ]. For each i note that fi γ : R → X is
a quasi-axis for fi hi fi−1 . After extracting a subsequence of fi we may assume that
the following three things hold. First, if i 6= j then fi (∂± h) 6= fj (∂± h), and hence
fi−1 fj 6∈ Stab(∂± h), and so after deleting at most a single term of the sequence
we have fi 6∈ Stab(∂± h) for all i. Second (and after possibly inverting h) we have
fi (∂+ h) 6= ∂+ h and fi (∂+ h) 6= ∂− h. Third, there are integer sequences ai → −∞,
bi → +∞, and ci < di such that d(γ(ai T ), fi γ(ci T )) ≤ L and d(γ(bi T ), fi γ(di T )) ≤
L + ∆ = L1 . Letting gi = h−ai fi hci , and letting Bi = bi − ai and Di = di − ci ,
we have d γ(0), gi (γ(0)) ≤ L1 and d γ(Bi T ), gi γ(Di T )) ≤ L1 , and we also have
Bi , Di → +∞. We shall prove that some infinite subsequence of gi is pairwise distinct
and satisfies Definition 2.4 (1) and (2) and hence h does not satisfy WWPD.

For proving that Definition 2.4 (1) holds, we claim that d γ(Di T ), γ(Bi T ) ≤
4L + 2∆, from which we obtain the bound




d γ(Di T ), gi (γ(Di T )) ≤ d γ(Di T ), γ(Bi T ) + d γ(Bi T ), gi γ(Di T ) ≤ 5L + 3∆
|
{z
} |
{z
}
≤4L+2∆

≤L+∆

We prove the claim assuming Bi ≥ Di ; the other case is similar. The point γ(Di T ),
which lies on the k, c quasi-geodesic path γ[0, Bi T ], is within distance L of some point

14

P in a geodesic [γ(0), γ(Bi T )]. We have
d(γ(0), γ(Di T )) + d(γ(Di T ), γ(Bi T ))
≤ d(γ(0), P ) + d(P, γ(Di T )) + d(γ(Di T ), P ) + d(P, γ(Bi T ))
= d(γ(0), γ(Bi T )) + 2L

≤ d gi (γ(0)), gi (γ(Di T )) + 2L + 2L1
= d(γ(0), γ(Di T )) + 4L + 2∆
from which the claim follows.
We next prove, after passing to a subsequence, that for all i 6= j we have
gi−1 gj 6∈ Stab(∂+ h)

(∗)

implying pairwise distinctness of the gi , and implying that gi−1 gj 6∈ Stab(∂± h) and so
Definition 2.4 (2) holds. Fixing j it suffices to prove that (*) holds for all sufficiently
large i, and so consider those values of i such that the following hold:
gi−1gj = h−ci fi−1 hai −aj fj hci ∈ Stab(∂+ h)
fi−1 hai −aj fj ∈ Stab(∂+ h)
hai −aj (fj (∂+ h)) = fi (∂+ h)

(∗∗)

On the left hand side of (∗∗), since fj (∂+ h) 6= ∂+ h, and since ai − aj → −∞ as
i → +∞, it follows by source-sink dynamics that hai −aj (fj (∂+ h)) → ∂− h. But on the
right hand side we have fi (∂+ h) → ∂+ h. Equation (∗∗) is therefore impossible for all
sufficiently large i.
This completes the proof of equivalence of items (1), (2), (3), (4)∀ and (4)∃ .
To prove (5) it suffices by symmetry it suffices to prove that Stab(∂− h) = Stab(∂± h).
For that it suffices, given γ such that γ(∂− h) = ∂− h, to prove that γ(∂+ h) = ∂+ h.
If not then ∂− (γhγ −1 ) = γ(∂− h) = ∂− h and ∂+ (γhγ −1 ) = γ(∂+ h) 6= ∂+ h. Using
source sink dynamics of h acting on ∂X it follows that the sequence hn (∂± (γhγ −1 )) =
∂± ((hn γ)h(hn γ)−1 ) converges to ∂± h as n → +∞, but the terms of that sequence are
all distinct from ∂± h, contradicting that h has equivariantly discrete fixed points.
To prove (6), consider k ∈ Γ−Stab(∂h) and suppose that khk −1 is not independent
of h, so the set ∂h ∩ k(∂h) = ∂h ∩ ∂(khk −1 ) consists of a single point. If ∂− h =
k(∂− h) or if ∂+ h = k(∂+ h) then either of the elements k or khk −1 gives an immediate
contradiction to (5). If ∂− h = k(∂+ h) = ∂+ (khk −1 ) = ∂− (kh−1 k −1 ) then kh−1 k −1
gives a contradiction to (5), and similarly if ∂+ h = k(∂− h).

2.3

Equivalence g ∼ h in a hyperbolic group action

Given two paths γ : [a, b] → X and ρ : [c, d] → X in a metric space X, we say that
γ, ρ are L-Hausdorff close rel endpoints if d(γ(a), ρ(c)) ≤ L, d(γ(b), ρ(d)) ≤ L, and
the Hausdorff distance between the sets γ[a, b] and ρ[c, d] is ≤ L.
15

The following relation is defined for loxodromic elements by Bestvina and Fujiwara
[BF02]. We trivially extend the relation by convention to nonloxodromic elements,
with the effect that if g is not loxodromic and g ∼ h then g = h.
Definition 2.9. Define an equivalence relation g ∼ h on Γ to mean: either g = h; or
g, h are loxodromic, and for some (any) quasi-axes γg and γh there exists L such that
for any subsegment γg [a, b] there is a subsegment γh [c, d] and an element k ∈ Γ such
that k(γg [a, b]) is L-Hausdorff close rel endpoints to γh [c, d].
For each g ∈ Γ the orbit of ∂± g under the action Γ y ∂S × ∂S − ∆ is denoted
Γ · ∂± g, and its closure in ∂S × ∂S − ∆ is denoted Γ · ∂± g.
Lemma 2.10. Given loxodromic g, h ∈ Γ, the following are equivalent:
(1) g ∼ h
(2) Γ · ∂± g = Γ · ∂± h
(3) Γ · ∂± g ∩ Γ · ∂± h 6= ∅
If in addition g satisfies WWPD then the following condition is also equivalent:
(4) Γ · ∂± g = Γ · ∂± h.
Proof. Evidently (2) =⇒ (3). Pick quasi-axes γg of g and γh of h.
To prove (1) =⇒ (2), assuming g ∼ h, for each n there exists αn ∈ Γ such
that γg [−n, +n] is L-Hausdorff close rel endpoints to some subsegment of αn (γh ) =
, and it follows that ∂± (αn hαn−1 ) → ∂± g as n → +∞. This shows that
γαn hα−1
n
∂± g ∈ Γ · ∂± h, and so the inclusion Γ · ∂± g ⊂ Γ · ∂± h holds. The opposite inclusion
holds by symmetry.
To prove (3) =⇒ (1), choosing (ξ, η) ∈ Γ · ∂± g ∩ Γ · ∂± h, choose sequences αn , βn
in Γ such that the sequences αn (∂g) = ∂(αn gαn−1) and βn (∂g) = ∂(βn hβn−1 ) both
approach (ξ, η). Choose an oriented quasigeodesic δ with initial end ξ and terminal
end η. By precomposing both αn and βn with appropriate powers of g we may
assume that αn (γg (0)) and βn (γh (0)) each stay within a uniform distance of δ(0).
After passing to subsequences of αn and βn it follows there is a constant L and
−
+ +
− +
− +
sequences s−
n , tn → −∞ and sn , tn → +∞ such that α(γg [sn , sn ]) and β(γh [tn , tn ])
are each L/2-Hausdorff close rel endpoints to δ[−n, +n], and so are L Hausdorff close
+
to each other. Since the paths γg [s−
n , sn ] exhaust the quasi-axis γg , as do the paths
+
γh [t−
n , tn ], it follows that g ∼ h.
Clearly (4) =⇒ (2). For the converse, assuming g satisfies WWPD then by discreteness of Γ · ∂± g it follows that Γ · ∂± g = Γ · ∂± g. Assuming (2) we therefore have
Γ · ∂± h = Γ · ∂± g, and so Γ · ∂± h ⊂ Γ · ∂± g. By transitivity of the Γ-action on Γ · ∂± g
it follows that Γ · ∂± g = Γ · ∂± h which is (4).

16

Remark. Note that if two loxodromics g, h ∈ Γ satisfy ∂− g = ∂− h then g ∼ h
because limi→+∞ g i (∂± h) = ∂± g and one may apply Lemma 2.10 (3) =⇒ (1); similarly
if ∂+ g = ∂+ h. On the other hand if ∂− g = ∂+ h, or even if h = g −1, one cannot in
general conclude anything about whether g ∼ h.

2.4

Individualized WWPD methods.

We recall the main outline of [BF02], which gives a method for using a group action
Γ y S on a hyperbolic complex to compute second bounded cohomology.
[BF02] Theorem 1: If the action has an independent pair of loxodromic elements
g, h ∈ Γ such that g 6∼ h, then Hb2 (Γ; R) is of uncountably infinite dimension.
[BF02] Proposition 6 (5): If the action Γ y S is WPD then Γ has independent
loxodromic elements g 6∼ h.
Putting these together one obtains:
[BF02] Theorem 7: If the action Γ y S is WPD then Hb2 (Γ; R) is of uncountably
infinite dimension.
An important feature of the proof of [BF02] Proposition 6 (5) is that it works under
weaker hypotheses, namely that some loxodromic g ∈ Γ satisfies the individualized
WPD property and that the action Γ y S has an independent pair of loxodromic
elements. The proof of [BF02] Theorem 7 therefore also works under the same weaker
hypotheses.
What follows in Proposition 2.11 is a version of [BF02] Proposition 6 based on
the WWPD property for an individual element. Proposition 2.11 also contains some
additional details needed for later use.
We first review the basic facts about hyperbolic ping-pong (see e.g. [Mos07, Theorem 6.7]). Consider a geodesic hyperbolic space S. Recall that a group action F y S
is Schottky if F is free of finite rank and for some (any) y ∈ S the orbit map F 7→ S
given by f 7→ f · y is a quasi-isometric embedding with respect to the word metric on F ; in particular each nontrivial element of F is loxodromic. Recall also the
following basic facts:
The Cayley tree of a Schottky group: For any Schottky action F y S, and for
any free basis of F with associated Cayley tree T , there exists an F -equivariant
continuous quasi-isometric embedding T 7→ Y which has a unique continuous
extension to the Gromov bordifications T ∪ ∂T 7→ Y ∪ ∂Y .
Hyperbolic ping pong: For any pair of independent loxodromic isometries α, β ∈
Isom(S), and any pairwise disjoint neighborhoods U− , U+ , V− , V+ ⊂ ∂S of
∂− α, ∂+ α, ∂− β, ∂+ β there exists an integer M ≥ 1 such
 that for anymm,nn ≥ M
n
m
the following hold: α ∂S − U− ⊂ U+ , β ∂S − V− ⊂ V+ , and α , β freely
generate a Schottky action F (αm , β n ) y S.
17

Proposition 2.11. Let γ ∈ Γ satisfy WWPD and suppose that Γ 6= Stab(∂γ). For
any α ∈ Γ − Stab(∂γ) there exists A > 0 such that for any a ≥ A, letting g1 = γ a and
h1 = γ a (αγα−1 )−a = γ a αγ −a α−1 , we have:
(1) h1 is also loxodromic
(2) g1 6∼ h1 and g1 6∼ h−1
1
(3) g1 , h1 freely generate a Schottky subgroup of Γ.
Proof. Consider the loxodromic δ = αγα−1. Applying Proposition 2.6 (6), δ and γ
are independent. Now we play hyperbolic ping-pong with some WWPD spin. Choose
pairwise disjoint neighborhoods U− , U+ , V− , V+ ⊂ ∂S ×∂S −∆ of ∂− γ, ∂+ γ, ∂− δ, ∂+ δ,
respectively, where U− , U+ are chosen so small that ∂± γ is the only element of the
Γ-orbit of the ordered pair ∂± γ that is contained in U− × U+ . Choose M sufficiently
large so that for m, n ≥ M the conclusions of hyperbolic ping pong hold as stated
above, in particular γ m , δ n freely generate a Schottky action on S. For a ≥ A = 2M,
letting g1 = γ a and h1 = γ a δ −a = γ a αγ −a α−1 , it follows that h1 is loxodromic and
that the two elements g1 , h1 freely generate a Schottky subgroup, because any finitely
generated subgroup of the Schottky group hγ a , δ a i is also Schottky.
Having verified conclusions (1), (3) of Proposition 2.11, we turn to (2).
Letting U+′ = γ M (U+ ) ⊂ U+ , we have ∂− h1 ∈ V+ , ∂+ h1 ∈ U+′ , and γ −M (∂± h1 ) =
∂± (γ −M h1 γ M ) ∈ U− × U+ . But γ −M h1 γ M is independent of γ and so γ −M (∂± h1 ) 6=
∂± γ. From our choice of U− × U+ and the fact that γ satisfies WWPD, by applying
Lemma 2.10 (4) it follows that γ 6∼ γ −M h1 γ M . Since 6∼ is invariant under conjugacy
and under passage to powers, it follows that g1 6∼ h1 .
The exact same argument (with α−1 in place of α, and with the inclusion γ −M (U− ) ⊂
U− in place of the inclusion γ M (U+ ) ⊂ U+ ) shows that g1 6∼ h′1 = γ a α−1 γ −a α. But
−1
since h′1 is conjugate to h−1
1 it follows that g1 6∼ h1 .
Remark. Putting together Proposition 2.11 and [BF02] Theorem 1 we immediately
obtain Theorem 2.12 below regarding group actions on hyperbolic spaces possessing
WWPD elements. One may sometimes use Theorem 2.12 in place of Theorem D,
perhaps at the expense of weaker conclusions. For example, using Theorem 2.12
it follows that a group satisfying the hypotheses of Theorem E and whose action is
nonelementary has an embedded ℓ1 . Also, the reduction arguments of Section 4.3 may
be adapted, and combined with Theorem 2.12, to prove a weak version of Theorem A
saying that any finitely generated subgroup of Out(Fn ) has a finite index subgroup
satisfying the Hb2 -alternative.
Theorem 2.12. If Γ y S is a hyperbolic action possessing an independent pair of
loxodromic elements, and if Γ has a WWPD element, then Hb2 (Γ; R) contains an
embedded ℓ1 .

18

2.5

Global WWPD methods: Statement of Theorem D.

In [BF02] Theorem 8, Bestvina and Fujiwara generalize their Theorem 7 by giving
hypotheses on a hyperbolic action of a finite index subgroup N of a group Γ which are
sufficient to determine Hb2 (Γ; R). The hypotheses require that the subgroup action
N y X satisfies the global WPD property, and that there is an embedding of Γ into
a certain wreath product which interacts with the subgroup action in a certain way.
Those hypotheses are tailored to the case where Γ is a subgroup of a surface mapping
class groups, allowing one to apply Ivanov’s subgroup decomposition theory to obtain
a natural finite index subgroup of Γ acting on a certain subsurface curve complex, and
allowing one to use the theorem of [BF02] saying that every element of the mapping
class group that acts loxodromically on the curve complex satisfies WPD.
Theorem D (aka Theorem 2.5) generalizes [BF02] Theorem 8 in a few ways. First,
WPD can be replaced by WWPD. Second, one does not need any wreath product
hypothesis at all: instead, one applies the Kalujnin-Krasner embedding of Γ into the
wreath product of N by Γ/N. Third, when we come to apply Theorem D in the
setting of Out(Fn ), we will not know that every loxodromic element of the action
satisfies WWPD, but we will know that the presence of one WWPD element implies
an entire Schottky subgroup’s worth of WWPD elements. The definition of the global
WWPD property therefore, by necessity, has a somewhat technical formulation, which
is designed to allow us to jump into the middle of the proof [BF02] Theorem 8 where
a certain Schottky subgroup of the action begins to play a significant role.
Definition 2.13. A group Γ is said to satisfy the global WWPD hypothesis if there
exists a normal subgroup N ⊳ Γ of finite index, an action N y X on a hyperbolic
complex, and a rank 2 free subgroup F < N, such that the following hold:
(1) Each element of N acts either loxodromically or elliptically on X.
(2) The restricted action F y X is Schottky and each of its nonidentity elements
is WWPD with respect to the action N y X.
(3) For each inner automorphism ig : Γ → Γ, letting N yg X denote the composed
ig

→ N y X, the restricted action F yg X satisfies one of the following:
action N −
(a) F yg X is Schottky and each of its nonidentity elements is WWPD with
respect to the action N yg X; or
(b) F yg X is elliptic.
In situations where N, its action N y X, and/or F are specified, we shall adopt
phrases like “Γ satisfies WWPD with respect to N, its action N y X, and/or F ”.
In practice we reformulate item (3) by using that N has finite index in order to
cut down the inner automorphisms that must be checked to a finite subset. Since
N is normal in Γ, the action of Γ on itself by inner automorphisms restricts to an
19

action of Γ on N that we denote i : Γ → Aut(N). Choose coset representatives gκ of
N in Γ, where κ ∈ {1, . . . , K} and K = [Γ : N], and by convention choose g1 to be
the identity. Let iκ ∈ Aut(N) be the restriction to N of igκ ∈ Inn(Γ). We refer to
i1 , . . . , iK as outer representatives of the action of Γ on N (see diagram below). Let
iκ
N yκ X be the composed action N −
→
N y X, and let F yκ X be its restriction.
With this notation, to verify (3) as stated for all the actions F yg X it suffices to
check only the actions F yκ X. In other words, (3) is equivalent to:
(3)′ For each κ = 1, . . . , K the action F yκ X satisfies one of the following:
(a) F yκ X is Schottky and each of its nonidentity elements is WWPD with
respect to the action N yκ X; or
(b) F yκ X is elliptic.
The terminology of “outer representatives” refers to the fact that in the commutative
diagram
i
/ Aut(N)
Γ


Γ/N


/

Out(N)

Aut(N)/ Inn(N)

the automorphisms iκ ∈ Aut(N) represent all of the elements of the image of the
homomorphism Γ/N → Out(N) (that homomorphism need not be injective, and
so there may be some duplication of elements of Out(N) represented by the list
i1 , . . . , iK , but this is inconsequential). The equivalence of (3) and (3)′ holds because
if we replace gκ by something else h = νgκ in its coset (ν ∈ N), then iκ ∈ Aut(N)
is replaced by ih = iν ◦ iκ ∈ Aut(N), and so the restricted actions F yκ X and
F yh X are conjugate by an isometry of X, namely the action of ν ∈ N. But each
of properties (a) and (b) is invariant under such conjugation.
This completes Definition 2.13.
Remark. We emphasize that although the various actions N yκ X are equivalent
up to inner automorphisms of Γ restricted to its normal subgroup N, that does not
mean that they are equivalent up to conjugation by isometries of X, because Γ itself
does not act on X. Thus, for example, an element of N may be loxodromic with
respect to one of the actions N yκ X but not with respect to a different one.
The following lemma explains how the global WWPD hypothesis is a “fragment”
of the WPD property, and it gives more information we shall use elsewhere.
Lemma 2.14. Let A : Γ → Isom(X) be an action on a hyperbolic complex, and let
Q = Image(A). If Q satisfies WPD, and if each element of Q is either loxodromic
or elliptic, then the given action A of the group Γ on X satisfies the global WWPD
hypothesis with respect to the N = Γ.
20

Proof. Applying [BF02] Proposition 6 (4) to Q we obtain independent loxodromic
elements of Q, and by applying hyperbolic ping-pong we obtain a rank 2 Schottky
subgroup F ′ < Q with respect to the action Q y X. Under the homomorphism
A : Γ → Q there is a section F ′ → Γ whose image F < Γ is a rank 2 Schottky subgroup
with respect to the action Γ y X. Since Q satisfies WPD, applying Corollary 2.8 it
follows that each nontrivial element of F satisfies WWPD with respect to the action
Γ y X.
Here, restated from the introduction, is our main result regarding the global
WWPD hypothesis; see the appendix for the proof.
Theorem D. If a group Γ satisfies the global WWPD hypothesis then Hb2 (Γ; R) contains an embedding of ℓ1 .
When we apply Theorem D, the free subgroup implicit in the global WWPD
hypothesis will be constructed by ping-pong methods.

3

Background material on Out(Fn)

Section 3.1 reviews basic definitions, terminology, and facts regarding Out(Fn ), referring to the literature for full details.
Section 3.2 contains Lemma 3.1, a uniform splitting property which will be used
for our study of well functions in the proof of Proposition 5.7, and in Part II of this
work.

3.1

Basic concepts of Out(Fn)

In this subsection we briefly review basic terminology and notation and key definitions, focussing on relative train track maps. The original sources for this material
include [CV86], [BH92], [BFH00], and [FH11]. See [HM17c, Section 1] for full definitions and citations in a still brief but much more comprehensive overview.
Marked graphs and topological representatives. A rank n marked graph is
a finite graph G without valence 1 vertices equipped with a homotopy equivalence
Rn 7→ G, where Rn is the base rose of rank n whose edges are indexed and oriented
so as to identify Fn ≈ π1 (Rn ). The marking induces an isomorphism Fn ≈ π1 (G),
e on the universal cover, and an equivariant
a deck transformation action Fn y G
e
homeomorphism ∂Fn ≈ ∂ G, all well-defined up to inner automorphism of Fn . A
homotopy equivalence f : G → G′ of marked graphs is always assumed to take vertices
to vertices and to be locally injective on each edge, and so f induces a map of
directions (initial germs of oriented edges) and a map of turns (pairs of directions
at the same vertex). Given a marked graph G there is an induced isomorphism
between the group of self-homotopy equivalences of G modulo homotopy and the
group Out(Fn ). A self-homotopy equivalence f : G → G that represents φ ∈ Out(Fn )
21

in this way is called a topological representative of φ; in such a case, iterating the
induced map on turns, a turn is illegal if some iterate is degenerate, and it is legal
otherwise.
Paths, lines, and circuits. Following [BFH00] a path in a marked graph G is
a concatenation of edges and partial edges without backtracking, such that a partial
edge may occur only incident to an endpoint of the path; in the degenerate case,
a trivial path consists of just a vertex. An infinite path in G can be a line or a
ray depending on whether the concatenation is doubly infinite or singly infinite. In
some situations we must restrict to paths with endpoints at vertices, e.g. for defining
complete splitting of paths; in other situations we must allow for more general paths
with arbitrary endpoints, e.g. when taking pre-images of paths. In other situations
we will use more general edge-paths in which backtracking is allowed. We will use
appropriate language to indicate such situations.
A circuit is a cyclic concatenation of edges without backtracking, i.e. a circle
immersion.
Two paths or circuits are equivalent if they differ only by reparameterizaiton and
b
inversion. The space of paths and circuits up to equivalence is denoted B(G),
and
is equipped with the weak topology having a basis element for each finite path α,
consisting of all paths and circuits having α as a subpath. The subspace of lines is
b
denoted B(G) ⊂ B(G).
The abstract space of lines of the group Fn is the quotient
e
e n ) is the space of two point subsets of
space B = B(Fn ) = B(Fn )/Fn where B(F
∂Fn , endowed with the weak topology naturally induced by the topology of ∂Fn . The
e induces a canonical homeomorphism B(Fn ) ≈ B(G). The
identification ∂Fn ≈ ∂ G
element of B(G) corresponding to a given ℓ ∈ B(Fn ) is called the realization of ℓ in G.
The natural action Aut(Fn ) y ∂Fn induces an action Out(Fn ) y B.
Path maps. [HM17c, Section 1.1.6]. Consider a homotopy equivalence of possibly
distinct marked graphs f : G → G′ that represents φ ∈ Out(Fn ) in a manner similar
f
to a topological representative: the composition Rn 7→ G −
→ G′ 7→ Rn defined using
marking maps or their homotopy inverses is homotopic to a topological representative
b
b ′ ), where f# (γ) is obtained from
of φ. There is an induced path map f# : B(G)
→ B(G
the composition f ◦ γ by straightening rel endpoints, lifting first to the universal cover
when γ is infinite in order to make use of endpoints at infinity. For each line in B
realized as ℓ ∈ B(G) its image φ(ℓ) ∈ B is realized as f# (ℓ) ∈ B(G′ ).
b
b ′ ) defined as follows: for any
There is another induced path map f## : B(G)
→ B(G
b
e choose any lift f˜: G
e→G
e′ ,
path γ ∈ B(G),
lift to a path γ̃ in the universal cover G,
e′ that contain γ̃ as a subpath, and
let f˜## (γ̃) be the intersection of all paths in G
′
let f## (γ) be the projection to G of f˜## (γ̃), which is well-defined independent of
choices. This path f## (γ) is obtained from f# (γ) by truncating a subpath incident
to each endpoint of length at most equal to the bounded cancellation constant of f .
Assuming now that G = G′ , a finite path γ is a Nielsen path if f# (γ) = γ. If a
Nielsen path can not be written as a concatenation of two non-trivial Nielsen paths
22

then it is indivisible.
A decomposition γ = γ1 · . . . · γK of a finite path into subpaths is a splitting if
i
f# (γ) = f#i (γ1 ) . . . f#i (γK ) for all integers i ≥ 1 (the single dot · in expressions like
γ1 · . . . · γK always refers to a splitting). If P is a set of finite paths and if each γi ∈ P
then we say that γ = γ1 · . . . · γK is a splitting of γ with terms in P . This concept
of “splitting with terms in P ” is most useful if for each α ∈ P the path f# (α) splits
with terms in P ; see for example the definition of a CT.
Free factor systems and free factor supports. [BFH00]. Conjugacy classes
in Fn are denoted [·]. A subgroup system in Fn is a set A = {[A1 ], . . . , [AK ]} such that
A1 , . . . , AK < Fn are nontrivial finitely generated subgroups [HM17c, Section 1.1.2].
The elements of a subgroup system are called its components. The partial ordering
of containment A ⊏ A′ means that for all [A] ∈ A there exists [A′ ] ∈ A′ such that
A < A′ .
A subgroup system A is a free factor system if it has the form A = {[A1 ], . . . , [AK ]}
so that there is a free factorization Fn = A1 ∗ · · · ∗ AK ∗ B. Associated to each
subgraph H ⊂ G of a marked graph is the free factor system [H] = [π1 H] =
{[π1 H1 ], . . . , [π1 HK ]}, where H1 , . . . , HK are the noncontractible components of H and
[π1 Hk ] denotes the conjugacy class of the image of the inclusion induced monomorphism π1 (Hk ) ֒→ π1 (G) ≈ Fn . Every chain A1 ⊏ · · · ⊏ AL of free factor systems can
be realized in this manner as a filtration G1 ⊂ · · · ⊂ GL ⊂ G of subgraphs of some
marked graph G.
A line ℓ ∈ B(Fn ) is supported by or carried by a subgroup system A if there exists
e n ) covering ℓ and A < Fn such that [A] ∈ A and such that ∂ ℓ̃ ⊂ ∂A; let
ℓ̃ ∈ B(F
B(A) = {ℓ ∈ B(Fn ) ℓ is carried by A}; equivalently, if A is realized by a subgraph
H ⊂ G then the realization of ℓ in G is contained in H.
The free factor support of a subset L ⊂ B(Fn ), denoted Fsupp (L), is the unique
minimal free factor system Fsupp (L) which supports each element of L. Also, given a
subgroup system A we use the shorthand notation Fsupp (L; A) = Fsupp (L ∪ B(A)), and
we call this the free factor support of L relative to A. Also, we say that L fills relative
to A if Fsupp (L; A) = {[Fn ]}.
Relative train track maps. [BH92]. A topological representative f : G → G
is filtered if it comes equipped with an f -invariant filtration of subgraphs ∅ = G0 ⊂
G1 ⊂ · · · ⊂ GR = G, with corresponding subgraphs Hr = Gk \ Gr−1 called strata,
such that if Hr has m edges with corresponding m × m transition matrix M then
either Hr is an irreducible stratum meaning that M is irreducible or Hr is a zero
stratum meaning that M is a zero matrix. Irreducible strata Hr are further classified
by the value of the Perron-Frobenius eigenvalue λ ≥ 1 of M: Hr is an EG-stratum
if λ > 1; and Hr is an NEG-stratum if λ = 1. An EG stratum is aperiodic if M is
Perron-Frobenius (some power of M is positive).
For f : G → G to be a relative train track representative requires some conditions
to be imposed on its EG strata. The definition and some further properties are found
in [BH92] Section 5, including a reformulation of the definition saying that for each
23

EG stratum Hr the following must hold:
(RTT-(i)) Each initial direction of an edge in Hr maps to some initial direction of
an edge in Hr .
Paths in Gr whose only turns in Hr are legal turns are called Hr -legal paths. An edge
in Hr , having no turns at all, is by default Hr -legal.
([BH92] Lemma 5.8) If σ is an r-legal path in Gr with endpoints in Hr then σ
splits with terms in
{edges of Hr } ∪ {nontrivial subpaths of Gr−1 with endpoints in Hr }
and f# (σ) is r-legal.
Paths of the form f#k (E), for k ≥ 0 and edges E ⊂ Hr , are called k-tiles of Hr (note
that when f k (E) is straightened to form f#k (E), no edges of Hr are cancelled).
Every φ ∈ Out(Fn ) has a relative train track representative [BH92]. Furthermore,
some positive power φk has a relative train track representative f : G → G which is
EG-aperiodic, meaning that the transition matrix of each EG stratum is a PerronFrobenius matrix; one may take f to be the straightened k th power of any relative
train track representative of φ itself, with a refined filtration.
Attracting Laminations. [BFH00]. Consider φ ∈ Out(Fn ) and an EG-aperiodic
relative train track representative f : G → G of some positive power φi . Associated
to each EG stratum Hr ⊂ G is its attracting lamination Λr , a closed subset of B(Fn )
whose realization in G consists of all lines ℓ such that each subpath of ℓ is contained in
b
some k-tile of Hr ; equivalently, the lines of Λr are the weak limits in B(G)
of sequences
of k-tiles. The lamination Λr is also characterized as the closure of some birecurrent,
nonperiodic line ℓ ∈ B ≈ B(G) of height r such that for some weak neighborhood
U ⊂ B of ℓ we have φ(U) ⊂ U and {φi# (U) i ≥ 0} is a neighborhood basis of ℓ; each
such line ℓ is in Λr , ℓ is called a generic leaf of Λr , and each such neighborhood U is an
attracting neighborhood of a generic leaf. The set L(φ) = {Λr Hr is an EG stratum}
is well-defined independent of the choice of f : G → G, and it is a finite set.
The elements of the set L(φ) are distinguished by their free factor supports: if
Λ1 6= Λ2 ∈ L(φ) then Fsupp (Λ1 ) 6= Fsupp (Λ2 ). The elements of the two sets L(φ) and
L(φ−1 ) correspond one-to-one according to their free factor supports: the relation
Fsupp (Λ+ ) = Fsupp (Λ− ) between Λ+ ∈ L(φ) and Λ− ∈ L(φ−1 ) is a bijection. We will
refer to the ordered pair Λ± = (Λ+ , Λ− ) as a (dual) lamination pair of φ, and we
write their common support as Fsupp (Λ± ). The set of all lamination pairs of φ is
denoted L± (φ).
Given a lamination pair Λ± ∈ L± (φ) which is fixed by φ, a line ℓ ∈ B is weakly
attracted to Λ+ (by iteration of φ) if φk (ℓ) weakly converges to a generic leaf of Λ+ .
More details of weak attraction are found in Section 4.1.

24

EG properties of CTs. There is a particularly nice kind of relative train track
map called a CT. The necessary and sufficient condition for φ to be represented by
a CT is that φ be rotationless [FH11, Definition 3.13]. Every φ has a rotationless
iterate with uniformly bounded exponent [FH11, Theorem 4.28] so one can often
work exclusively with CTs. The defining conditions for f to be a CT are found
in [FH11, Definition 4.7], a sequence of nine conditions with parenthesized names
(some mentioned here). Most of our applications will be to a CT f : G → G with an
f -invariant filtration ∅ = G0 ⊂ G1 ⊂ · · · ⊂ Gs = G satisfying the following:
• (Special Assumption) The top stratum Hs is EG-aperiodic.
For now, instead of a full and general definition, we only list certain properties of
a CT f : G → G which hold under the special assumption above (citations are to
[FH11] unless otherwise noted):
(1) (Definition 4.7 (Filtration)) There is no φ-periodic free factor system strictly
contained between the free factor systems [π1 Gs−1 ] and [π1 Gs ] = {[Fn ]}.
(2) (Corollary 4.19 eg-(i)) There exists (up to reversal) at most one indivisible
Nielsen path ρ of height s. If ρ exists then it decomposes as ρ = αβ where α
and β are s-legal paths with endpoints at vertices and (ᾱ, β) is an illegal turn
in Hs . In particular, f (α) = αγ and f (β) = γ̄β for some path γ. Moreover:
• ([HM17c, Fact 1.42]) At least one endpoint of ρ is disjoint from Gs−1 .
• (Corollary 4.19 eg-(i)) The initial and terminal directions of ρ are distinct
fixed directions in Hs .
(3) ([BH92, Lemma 5.10]) Any assignment ℓ(E) of lengths to the edges e of Hs
extends to an assignment of lengths to all paths in G with endpoints at vertices
by adding up the lengths of the edges of Hs in the edge-path description of σ.
There is an eigenvector assignment of lengths ℓ(e) so that ℓ(f (e)) = λℓ(e) for all
edges e of Hs where λ > 1 is the Perron-Frobenius eigenvalue of the transition
matrix for Hs . If ρ = αβ exists as in (2) it follows that ℓ(α) = ℓ(β) because
ℓ(α) and ℓ(β) both satisfy λL = L + ℓ(γ).
(4) ([HM17c, Fact 1.35]) If γ is a path with endpoints at vertices Hs or a circuit
crossing an edge of Hs then for all sufficiently large i the path f#i (γ) has a
splitting with terms in the set {edges of Hs } ∪ {height s indivisible Nielsen
paths} ∪ {paths in Gs−1 with endpoints in Hs }.
Remark on item (4): This is implicit in [BFH00], e.g. Lemma 4.2.6, Remark 5.1.2,
and Step 2 of the proof of Proposition 6.0.4 of [BFH00].
EG strata are further classified into geometric and nongeometric strata, as defined
and studied in [BFH00, Sections 5.1, 5.3] and further studied in [HM17c]. Very
roughly speaking Hs is geometric if the action of f on G may be modeled up to
homotopy by a self-homotopy-equivalence of a 2-complex that is obtained by attaching
25

a surface S to Gs−1 , such that on S the map is pseudo-Anosov. For now we need only
the following facts:
(5) ([HM17c, Fact 2.3]) Hs is geometric if and only if there is a closed height s
individual Nielsen path.
(6) ([HM17c, Section 2.4]) Geometricity is well-defined for a lamination pair Λ± ∈
L(φ) independent of the choice of a CT representative of a rotationless power
of φ. To be precise, if f : G → G and f ′ : G′ → G′ are CT representatives of
rotationless positive or negative powers of φ, with EG strata Hi ⊂ G, Hj′ ⊂ G′
corresponding to either Λ+ or Λ− depending on the sign of the power, then Hi ,
Hj′ are either both geometric or both nongeometric.

3.2

A uniform splitting lemma

In this section we consider a rotationless φ ∈ Out(Fn ) represented by a CT f : G → G
with an EG stratum Hs and a corresponding attracting/repelling lamination pair Λ±
s
each realized in G. If there exists an indivisible Nielsen path of height s (which must
be unique up to reversal), denote that path as ρ; otherwise, ignore ρ.
Lemma 3.1 will uniformize the splitting property that was reviewed in Section 3.1,
under item (4) of the heading “EG properties of CTs”. That property said that for
each finite path σ in G of height ≤ s having endpoints at vertices of Hs , some iterate
f#d (σ) splits into terms each of which is either an edge of Hs , an indivisible Nielsen
path of height s, or a path in Gs−1 with endpoints at vertices of Hs . The exponent d
needed for this splitting is unbounded in general, as one sees for example by letting
σ be a leaf segment of Λ−
s that crosses a large number of edges of Hs . The following
lemma says that such examples are the only reason for the splitting exponent d to be
unbounded.
For any path σ ⊂ G of height ≤ s, let |σ|s denote the number of times that σ
crosses edges of Hs . Also, let ℓ−
s (σ) denote the maximum of |τ |s over all paths τ in
G such that τ is a subpath both of σ and of a generic leaf of Λ−
s.
Lemma 3.1. With notation as above, for all L > 0 there is a positive integer d so that
if σ ⊂ G is a circuit or a finite path with endpoints at vertices of Hs , and if ℓ−
s (σ) < L,
then f#d (σ) splits into terms each of which is an edge of Hs , an indivisible Nielsen
path of height s, or a subpath of Gs−1 with endpoints at vertices; equivalently, f#d (σ)
splits into terms each of which is s-legal or an indivisible Nielsen path of height s.
Proof. We give the argument when σ is a finite path with endpoints at vertices of Hs ;
the argument for circuits is almost the same.
Arguing by contradiction, if the lemma fails then there exists L > 0 so that for
all even integers 2i there is a finite path σi ⊂ G with endpoints at vertices and with
2i
ℓ−
s (σi ) < L1 such that f# (σi ) does not split into terms each of which is either s-legal or
is an indivisible Nielsen path of height s. Let βi = f#i (σi ), and let βi = βi,1 ·. . .·βi,Ji be
26

a maximal splitting of βi into subpaths with endpoints at vertices. There is a splitting
σi = σi,1 · . . . · σi,Ji with f#i (σi,j ) = βi,j ; the paths σi,j need not have endpoints at
vertices.
If |βi,j |s is bounded independently of i and j then by [HM17c, Lemma 1.54] there
is a positive integer d independent of i, j such that each f#d (βi,j ) splits into subpaths
that are either s-legal or indivisible Nielsen paths of height s. The same is then true
for f#i (βi,j ) if i ≥ d, in contradiction to the fact that f#2i (σi ) = f#i (βi,1 ) · . . . · f#i (βi,Ji )
has no such splitting.
After passing to a subsequence we may therefore choose for each i an integer
j(i) ∈ {1, . . . , Ji } such that βi,j(i) s → ∞ as i → ∞. Since βi,j(i) does not split at any
interior vertex, there is a uniform bound to the number of edges of Hs in an s-legal
subpath βi,j(i) . It follows that the number of illegal turns of height s in βi,j(i) goes
to infinity with i. This in turn implies that σi,j(i) s goes to infinity with i, because
σi,j(i) has at least as many height s illegal turns as βi,j(i) .
It follows that some weak limit of the sequence σi,j(i) is a line of height s, contained
in Gs and containing at least one edge of Hs . Consider a height s line ℓ which is a weak
limit of a subsequence of σi,j(i) . If Hs is non-geometric then by the weak attraction
theory of [HM17e], the line ℓ has one of two options: either the weak closure of ℓ
contains Λ− ; or ℓ is weakly attracted to Λ+ [HM17e, Lemma 2.18]. If Hs is geometric
— and so ρ exists and is closed — then there is a third option [HM17e, Lemma 2.19],
namely that ℓ is a bi-infinite iterate of ρ or ρ̄.
Since no σi contains a subpath of Λ− that crosses L edges of Hs , neither does ℓ,
and so the weak closure of ℓ does not contain Λ− , ruling out the first option. Suppose
ℓ is weakly attracted to Λ+ . There exists m > 0 such that f#m (ℓ), and hence f#m (σi,j(i) )
for arbitrarily large i, contains an s-legal subpath that crosses > C edges of Hs , for
any choice of C. By [BFH00, Lemma 4.2.2], by choosing C to be sufficiently large we
may conclude that f#m (σi,j(i) ) splits at an interior vertex. But then for i ≥ m it follows
that βi,j(i) = f#i−m (f#m (σi,j(i) )) splits at an interior vertex, contradicting maximality
of the splitting of βi , and thus ruling out the second option for ℓ. This concludes the
proof if Hs is non-geometric.
Assuming now that Hs is geometric, it remains to show that the third option
can be avoided, and hence by the previous paragraph the desired contradiction is
achieved. That is, if Hs is geometric we show that some weak limit of a subsequence
of σi,j(i) contains at least one edge of Hs and is not a bi-infinite iterate of the closed
path ρ or ρ̄. This may be done by setting up an application of [HM17e, Lemma 1.11],
but it is just as simple to give a direct proof. Lift σi,j(i) to the universal cover of G
ei2 . . . E
eiM ⊂ G;
e the first and last terms
ei1 E
and write it as an edge path σ̃i,j(i) = E
i
are allowed to be partial edges. Let b equal twice the number of edges in ρs . Given
eim is well covered if there is a periodic
m ∈ {2 + b, . . . , Mi − b − 1}, we say that E
e that projects to ρ or to ρ̄ and that contains E
ei,m−b . . . E
eim . . . E
ei,m+b
line ρ̃im ⊂ G
as a subpath. Since the intersection of distinct periodic lines cannot contain two
eim
fundamental domains of both lines, ρ̃im is unique if it exists. Moreover, if both E
27

ei,m+1 are well covered then ρ̃im = ρ̃i,m+1 . It follows that if E
eim is well covered
and E
then we can inductively move forward and backward past other well covered edges
of σ̃i,j(i) all in the same lift of ρ, until either encountering an edge that is not well
covered, or encountering initial and terminal subsegments of σ̃i,j(i) of uniform length.
After passing to a subsequence, one of the following is therefore satisfied:
(1) There exists a sequence of integers Ki such that 2 + b < Ki < Mi − b − 1, and
eiK ⊂ H
e s is not well covered.
Ki → ∞, and Mi − Ki → ∞, and such that E
i

(2) σi,j(i) = αi ρpi βi where the number of edges crossed by αi and βi is bounded
independently of i and |pi | → ∞.

If (1) holds then the existence of a subsequential weak limit of σi,j(i) that crosses an
edge of Hs and is not a bi-infinite iterate of ρ or ρ̄ follows immediately. If (2) holds
then βi,j(i) = f#i (σi,j(i) ) decomposes as µi ρqi νi where |qi | → ∞ and the number of
illegal turns of height s in µi and νi is uniformly bounded. But then for any p ≥ 0
the number of edges of ρqi that are cancelled when f#p (µi )ρqi f#p (νi ) is tightened to
f#p (βi,j(i) ) is bounded independently of i and p and therefore βi,j(i) can be split at an
interior vertex, which is a contradiction.

4
4.1

Reducing Theorem A to Theorems C, D, E
Weak attraction and subgroup decomposition theory

In this section we review some terminology and notation regarding proofs and applications of lamination ping-pong, particularly from our papers on subgroup classification
theory [HM17c, HM17e, HM17f] and from Ghosh’s work on free subgroups of Out(Fn )
[Gho16].
The group IAn (Z/3). Recall the finite index normal subgroup IAn (Z/3) <
Out(Fn ) defined by

IAn (Z/3) = Ker Out(Fn ) 7→ Aut(H1 (Fn ; Z/3) ≈ GL(n, Z/3)
This subgroup has several important invariance properties:
[BT68] (and see [Vog02]) IAn (Z/3) is torsion free.
[HM17d, Theorem 3.1] For every ψ ∈ IAn (Z/3) and every free factor A < Fn , if its
conjugacy class [A] is ψ-periodic then it is fixed by ψ.
[HM17d, Theorem 4.1] For every ψ ∈ IAn (Z/3) and every γ ∈ Fn , if its conjugacy
class [γ] is ψ-periodic then it is fixed by ψ.
[HM17g] Every virtually abelian subgroup of IAn (Z/3) is abelian.
28

As said in the introduction, we sometimes emphasize the last result, from[HM17g], by
using the terminology “(virtually) abelian” in the context of a subgroup of IAn (Z/3).
For example, here is a consequence of [HM17g] which justifies that terminology for
the property of virtually abelian restrictions:
Corollary 4.1. A subgroup Γ < IAn (Z/3) has virtually abelian restrictions if and
only if it has abelian restrictions.
Proof. For any free factor A < Fn the inclusion A ֒→ Fn induces an injection
H1 (A; Z/3) 7→ H1 (Fn ; Z/3) which is equivariant with respect to the natural actions
Stab[A] ֒→ Out(Fn ) y H1 (Fn ; Z/3) and Stab[A] 7→ Out(A) y H1 (A; Z/3). It follows
that the image of Γ under the induced homomorphism Γ 7→ Out(A) is contained
in IAA (Z/3), and so by [HM17g] that image is virtually abelian if and only if it is
abelian.
Relative full irreducibility. An extension of free factor systems is simply an
instance F ⊏ F ′ of the usual partial ordering such that F =
6 F ′ . Given an extension
′
F ⊏ F , its co-edge number is the minimum, over all marked graphs G and subgraphs
H ⊂ H ′ ⊂ G realizing F ⊏ F ′ , of the number of edges of H ′ \ H. If the co-edge
number equals 1 then F ⊏ F ′ is a one-edge extension, and this holds if and only if the
free factor systems F and F ′ are related in one of two ways: either two components
of F having ranks i, j are replaced by a single component of F ′ of rank i + j and
containing the given two components of F ; or a single component of F having rank i is
replaced by a single component of F ′ of rank i+1 and containing the given component
of F . Otherwise F ⊏ F ′ is a multi-edge extension. If F ′ = {[Fn ]} then we drop F ′
from the terminology and speak of the co-edge number of F .
For any subgroup H < Out(Fn ) and any H-invariant extension F ⊏ F ′ , recall
that H is fully irreducible relative to the extension if there does not exist a free factor
system contained strictly between F and F ′ which is invariant under a finite index
subgroup of H. If in addition F ′ = {[Fn ]} then it is dropped from the terminology
and we say that H is fully irreducible rel F . When H = hφi is cyclic then we extend
this terminology to φ, and so φ is fully irreducible relative to a φ-invariant extension
F ⊏ F ′ if and only if no free factor system contained strictly between F and F ′ is
φ-periodic. In the context of a subgroup or element of IAn (Z/3) we may drop the
adverb “fully”, because in IAn (Z/3) periodic free factor systems are fixed [HM17d,
Theorem 3.1].
Weak attraction; nonattracting subgroup systems [HM17e, Section 1].
Consider a subgroup system A in Fn . We say that A is a vertex group system if there
exists a minimal action Fn y T on an R-tree T with trivial arc stabilizers such that
A is the set of conjugacy classes of nontrivial point stabilizers of the action Fn y T
[HM17c, Section 3].
±
Consider φ ∈ Out(Fn ) and a lamination pair Λ±
φ ∈ L (φ) which is fixed by φ.
Earlier we defined weak attraction of a line to the lamination Λ+
φ under iteration
29

of φ. A conjugacy class is weakly attracted to Λ+
φ if the periodic line representing
that conjugacy class is weakly attracted. The nonattracting subgroup system of Λ±
φ
±
is denoted Ana (Λφ ), it is concretely described in terms of any CT representative
of a rotationless power of φ in [HM17e, Definitions 1.2], and it is more abstractly
characterized by item (2) of the following compilation of results:
±
Fact 4.2. For any rotationless φ ∈ Out(Fn ) and Λ±
φ ∈ L (φ) we have:
±
(1) Ana Λ±
φ is a vertex group system, and it is a free factor system if and only if Λφ
is nongeometric ([HM17c, Section 3], [HM17e, Proposition 1.4]).

(2) Ana (Λ±
φ ) is the unique vertex group system such that: a conjugacy class c is
+
carried by Ana (Λ+
φ ) ⇐⇒ c is not weakly attracted to Λφ by iteration of φ
−1
[HM17e, Corollaries
⇐⇒ c is not weakly attracted to Λ−
φ by iteration of φ
1.9 and 1.10].
(3) If F is a proper, φ-invariant free factor system, and if there exists a lamination
±
pair Λ±
φ ∈ L (φ) not carried by F , then:
(a) F has co-edge number ≥ 2.
(b) If φ is fully irreducible rel F then Λ±
φ is the unique lamination pair in
±
L (φ) not carried by F , and furthermore:
±
i. Λ±
φ fills relative to F , meaning that Fsupp (Λφ ; F ) = {[Fn ]}.
±
ii. If Λ±
φ is nongeometric then Ana (Λφ ) = F ;

iii. If Λ±
φ is geometric then there exists a primitive cyclic subgroup C ⊂ Fn
such that Ana (Λ±
φ ) = F ∪{[C]} and Fsupp (F , [C]) = {[Fn ]}; in particular
[C] 6⊏ F .
Proof. Only item (3) needs some justification.
We prove (3a) by contradiction, so suppose that F has co-edge number 1. Let
f : G → G be a relative train track representative of φ in which F is represented by
a filtration element Gr . There are two cases to consider, regarding how G is obtained
topologically from Gr : by attaching either an arc or a “sewing needle” to Gr .
In the first case G is obtained by attaching an arc α to Gr , identifying each
endpoint of α with a vertex of Gr . Let f ′ : G → G be obtained from G by tightening
the restriction f α. Applying [BFH00, Corollary 3.2.2], the path f ′ (α) crosses α at
most once, and so f ′ has no EG stratum above Gr . Since f ′ Gr = f Gr , the map
f ′ is also a relative train track representative of φ, because the definition of a relative
train track map imposes no condition on non-EG strata. Applying [BFH00, Lemma
3.1.10], the highest stratum of f ′ crossed by leaves of Λ+
φ is EG and so all such leaves
are contained in Gr , implying that Λ+
is
supported
by
F , a contradiction.
φ
In the second case G is obtained from Gr by attaching a “sewing needle”, consisting
of a vertex v disjoint from Gr , an arc α with both endpoints identified to v, and
30

another arc α′ with one endpoint identified to v and opposite endpoint identified to
a vertex on Gr . Let f1 : G1 → G1 be the topological representative obtained from
f : G → G by first collapsing α′ to a point to get G1 and then tightening the induced
map on α. The analysis of the first case now applies to this map f1 , reaching the
same contradiction.
We prove (3b). For item (i), the free factor system Fsupp (Λ±
φ ; F ) equals {[Fn ]}
because it contains F and is φ-invariant, and φ is fully irreducible rel F . It also follows
that Hs is the top stratum. For uniqueness of Λ±
φ , if φ had a different lamination
pair not supported by F then there would exist a φ-invariant free factor system
strictly between F and {[Fn ]} (by [HM17c, Fact 1.55]), contradicting that φ is fully
irreducible rel F . The further conclusions of (3b) then follow from [HM17e, Definition
1.2, “Remark: The case of a top stratum”].
The conclusions of the following “mutual weak attraction” fact match key hypotheses of lamination ping-pong results, for example [HM17f, Proposition 1.3], and
Ghosh’s theorem [Gho16, Theorem 7.3] discussed below, enabling us to apply these
results below in Propositions 4.7 and 4.9.
±
±
±
Fact 4.3. Given φ, ψ ∈ Out(Fn ) and lamination pairs Λ±
φ ∈ L (φ) and Λψ ∈ L (ψ),
suppose that the following hypotheses hold:
+
−
+
(1) {Λ−
φ , Λφ } ∩ {Λψ , Λψ } = ∅;
±
+
(2) No generic leaves of Λ−
φ or Λφ are carried by Ana (Λψ ), and similarly with φ, ψ
switched;
+
−
It follows that generic leaves of Λ+
φ and of Λφ are weakly attracted to Λψ by iteration
−1
of ψ and to Λ−
ψ by iteration of ψ , and similarly with φ, ψ switched.
+
Proof. By symmetry we need only check that a generic leaf ℓ+
φ of Λφ is weakly attracted to Λ+
ψ by iteration of ψ, which we do by applying [HM17e, Theorem H].
+
−
−
Consider a generic leaf ℓ−
ψ of Λψ . If ℓψ is a leaf of Λφ then by combining [BFH00,
Lemma 3.1.15] with birecurrence of ℓ−
ψ it follows that one of two cases holds. In
+
−
−
the first case, ℓψ is a generic leaf of Λ+
φ and so Λψ = Λφ which contradicts (1). In
the second case, choosing any CT representative f : G → G of a rotationless power
of φ with core filtration element Gr and EG stratum Hr corresponding to Λ±
φ , the
±
−
−
realization in G of ℓψ is contained in Gr−1 and so ℓψ is carried by Ana (Λφ ), which
contradicts (2).
+
We conclude that ℓ−
ψ is not a leaf of Λφ and hence there exists a weak neigh+
−
+
borhood Vψ− of ℓ−
ψ such that ℓφ 6∈ Vψ . Also, from (2) we conclude that ℓφ is not
carried by Ana (Λ±
ψ ). Applying [HM17e, Theorem H (2)] it follows that for every weak
+
+
+
m +
neighborhood Vψ of ℓ+
ψ there exists m such that ψ (ℓφ ) ∈ Vψ , that is, ℓφ is weakly
attracted to ℓ+
ψ by iteration of ψ.

31

Lamination ping-pong. The next result is a pared down version of [HM17f,
Proposition 1.7], omitting several conclusions of that proposition labelled (2± ), (3± ),
(4± ) that we do not need here.
Proposition 4.4 (Proposition 1.7 of [HM17f]). Given a free factor system F , rota±
tionless φ, ψ ∈ Out(Fn ) both leaving F invariant, and lamination pairs Λ±
φ ∈ L (φ),
±
±1
±1
Λ±
(resp.) with fixed orientation,
ψ ∈ L (ψ) each having a generic leaf fixed by φ , ψ
assume that the following hypotheses hold:
±
(a) F ⊏ Ana Λ±
φ and F ⊏ Ana Λψ .
+
(i) Generic leaves of Λ+
ψ are weakly attracted to Λφ under iteration by φ.
−
−1
(ii) Generic leaves of Λ−
ψ are weakly attracted to Λφ under iteration by φ .
+
(iii) Generic leaves of Λ+
φ are weakly attracted to Λψ under iteration by ψ.
−
−1
(iv) Generic leaves of Λ−
φ are weakly attracted to Λψ under iteration by ψ .
Under these hypotheses, there exists an integer M, such that for any m, n ≥ M the
outer automorphism ξ = ψ m φn has an invariant attracting lamination Λ+
ξ ∈ L(ξ) and
−
−1
an invariant repelling lamination Λξ ∈ L(ξ ) such that each is nongeometric if Λ±
φ
±
and Λψ are nongeometric, and the following hold:
+
+
−
(1) F is carried by both Ana Λ−
ξ and Ana Λξ , and so neither Λξ nor Λξ is carried
+
±
±
by F . Also, both Ana Λ−
ξ and Ana Λξ are carried by Ana Λφ and by Ana Λψ .

(5+) For any weak neighborhood U + ⊂ B of a generic leaf of Λ+
ψ there exists an
+
+
+
integer M such that if m, n ≥ M then a generic leaf of Λξ is in U + .
(5-) For any weak neighborhood U − ⊂ B of a generic leaf of Λ−
φ there exists an
−
−
−
integer M such that if m, n ≥ M then a generic leaf of Λξ is in U − .
(6) Λ±
ξ is a dual lamination pair of ξ under either of the following conditions:
±
— Both pairs Λ±
φ and Λψ are nongeometric; or
−
— Both laminations Λ+
ξ and Λξ are geometric.

Subgroup decomposition theory. We review [HM17f, Theorem I] which is the
main result of subgroup decomposition theory. This theorem and its proof will be
applied in Proposition 4.6 to follow.
±
±
±
Consider φ, ψ ∈ Out(Fn ) and lamination pairs Λ±
φ ∈ L (φ), Λφ ∈ L (ψ). The In−
+
−
dependence Theorem [HM14b, Theorem 1.2] says that the sets {Λ+
φ , Λφ } and {Λψ , Λψ }
+
−
are either equal or disjoint. Furthermore Stab(Λ±
φ ) = Stab(Λφ ) = Stab(Λφ ), in other
−
+
−
words if θ ∈ Out(Fn ) stabilizes either of Λ+
φ or Λφ then θ stabilizes both Λφ and Λφ
[HM14b, Corollary 1.3].
Given φ ∈ Out(Fn ) and a φ-invariant free factor system F define
L(φ; F ) = {Λ ∈ L(φ) Λ is not carried by F}
32

and let L± (φ; F ) ⊂ L± (φ) be similarly defined. Given Γ < Out(Fn ) and an Γinvariant free factor system F , let L(Γ; F ) = ∪φ∈Γ L(φ; F ). Since free factor support
is well-defined for a dual lamination pair, the set L(Γ; F ) decomposes as a disjoint
union of dual lamination pairs.
Following [HM17f, Definition 1.2], we say that Γ is geometric above F if every
element of L(Γ; F ) is geometric.
Theorem 4.5 ([HM17f, Theorem I]). Let H < IAn (Z/3) be a subgroup and F ⊏
{[Fn ]} an H-invariant multi-edge extension of free factor systems such that H is
irreducible relative to F . If either H is finitely generated or L(H, F ) 6= ∅ then there
exists φ ∈ H which is fully irreducible relative to F . Moreover, for any θ ∈ H and
−
−1
Λ−
θ ∈ L(θ , F ), if either H is geometric above F or Λθ is non-geometric then for
any weak neighborhood U ⊂ B of a generic leaf of Λ−
θ we may choose φ so that generic
−1
leaves of Λ−
,
the
unique
element
of
L(φ
,
F
)),
are
contained in U.
φ
Laminations with infinite orbit. The following result will be used in Section 4.2. It is an application of [HM17f, Theorem I] and of the lamination ping-pong
tools underlying its proof.
Proposition 4.6. Consider a subgroup Γ < IAn (Z/3) and an Γ-invariant free factor
system F such that Γ is irreducible relative to F . If L(Γ; F ) contains more than one
dual lamination pair then for each Λ ∈ L(Γ; F ), the stabilizer of Λ in Γ has infinite
index, equivalently the Γ-orbit of Λ is infinite.
Proof. We begin with:
±
±
±
Claim 1: For any α, β ∈ Γ and lamination pairs Λ±
α ∈ L (α; F ), Λβ ∈ L (β; F ), if
−
±
+
α is fully irreducible rel F , and if {Λ−
6 {Λ−
α , Λα } =
β , Λβ }, then neither of Λβ is
−
a sublamination of either of Λ±
α , equivalently if γ is a generic leaf of one of Λβ
−
+
or Λ+
β then γ is a leaf of neither Λα nor Λα .
−
+
+
To prove this, note first that the sets {Λ−
α , Λα } and {Λβ , Λβ } are disjoint (by the Inde−
pendence Theorem [HM14b, Theorem 1.2]), so γ is a generic leaf of neither Λ+
β nor Λβ .
Also, by applying [BFH00, Lemma 3.1.15] to a relative train track representative of
α in which F is realized by a filtration element, it follows that every birecurrent,
nongeneric leaf of Λ±
α is carried by F and so no such leaf equals γ, completing the
proof of Claim 1.

Since L(Γ; F ) 6= ∅ it follows by Fact 4.2 (3) that F ⊏ {[Fn ]} is a multi-edge
extension. Applying Theorem 4.5, choose η ∈ Γ which is fully irreducible rel F . The
set L± (η; F ) consists of a unique lamination pair Λ±
η.
Let ΘL be the set of pairs (θ, Λ±
)
where
θ
∈
Γ, Λ±
θ
θ ∈ L(θ; F ) and the pairs
−
+
−
+
{Λθ , Λθ } and {Λη , Λη } are unequal and hence disjoint. By hypothesis, ΘL 6= ∅. For
±
±
each (θ, Λ±
θ ) ∈ ΘL , it follows by Claim 1 that generic leaves ℓθ of Λθ are neither
33

+
−
+
leaves of Λ−
η nor leaves of Λη . Since Λη and Λη are weakly closed, there are weak
±
±
−
neighborhoods Uθ ⊂ B of ℓθ disjoint from Λη and Λ+
η.
+
−
+
If Λ ∈ L(Γ; F ) is neither Λη nor Λη , choose (θ, Λ±
θ ) ∈ ΘL such that Λ = Λθ .
±
From Claim 1, Λ is not a sublamination of either of Λη . Applying [BFH00, Proposition 6.0.8], Λ is weakly attracted to Λ+
η under iteration of ξ. The η-orbit of Λ is there+
fore infinite, and so its Γ-orbit is infinite. This completes the proof if Λ 6∈ {Λ−
η , Λη }.
The same argument applies with η replaced by any other element of Γ that is irreducible rel F . We are therefore reduced to

Claim 2: There exists (µ, Λ±
µ ) ∈ ΘL such that µ is irreducible rel F .
±
As a first case, suppose that there exists (θ, Λ±
θ ) ∈ ΘL with non-geometric Λθ . By
Theorem 4.5 there exists µ ∈ Γ which is fully irreducible rel F and such that Λ−
µ is
−
−
−
+
contained in Uθ ; in particular, Λµ 6∈ {Λη , Λη } and we have verified claim 2. We are
±
now reduced to the case that each Λ±
θ is geometric. If Λη is geometric then the same
argument applies without change.
±
We may therefore assume that Λ±
η is non-geometric and that each Λθ is geometric.
±
±
Choose (µ, Λµ ) ∈ ΘL so that Ana Λµ is minimal with respect to ⊏. In other words,
±
±
±
±
±
if (θ, Λ±
θ ) ∈ ΘL and Ana Λθ ⊏ Ana Λµ then Ana Λθ = Ana Λµ . Such a (µ, Λµ ) exists
because there is a bound to the length of a chain of proper inclusions of non-attracting
subgroup systems [HM17c, Proposition 3.2]. We will complete the proof by showing
that µ is irreducible rel F . Since Λ±
µ is geometric, it suffices by [HM17f, Lemma 2.3
(3)] to show that StabΓ (Ana Λ±
)
has
finite index in Γ.
µ
±
Assuming that StabΓ (Ana Λµ ) has infinite index we will arrive at a contradiction
by applying Proposition 4.4. Passing to a power of µ we may assume that Λ±
µ have
generic leaves ℓ±
that
are
fixed
by
µ
with
fixed
orientations,
which
is
one
of the
µ
hypotheses of Proposition 4.4. Applying [HM17f, Lemma 2.1] there exists ζ ∈ Γ such
−
−1 +
that the following hold. None of the lines ζ(ℓ+
(ℓµ ), ζ −1(ℓ−
µ ), ζ(ℓµ ), ζ
µ ) is carried by
±
Ana Λµ ([HM17f, Lemma 2.1 (1)]), and hence by [HM17e, Corollary 2.17 (Theorem H)]
−1
each of these lines is weakly attracted to Λ±
µ under iteration by µ. Letting β = ζµζ
±
±
±
with geometric lamination pair Λβ = ζ(Λµ ) ∈ L (β; F ), it follows that µ and β
±
with their lamination pairs Λ±
µ and Λβ satisfy hypotheses (i)–(iv) of Proposition 4.4.
±
±
Also, Ana Λ±
β = ζ(Ana Λµ ) 6= Ana Λµ [HM17f, Lemma 2.1 (4)]. Hypothesis (a) of
±
±
±
Proposition 4.4 is that F ⊏ Ana Λ±
µ and F ⊏ Ana Λβ , which holds because Λµ , Λβ
are not supported by F . We may now apply Proposition 4.4 with U + = Uβ+ and
U − = Uµ− .
Applying the conclusions of Proposition 4.4, for some m, n ≥ 1 we have: an outer
automorphism ν = β m µn ; a ν-invariant attracting lamination Λ+
ν ∈ L(ν); and a ν+
invariant repelling lamination Λ−
∈
L(ν).
By
Conclusion
(1),
Λ−
ν
ν , Λν ∈ L(Γ; F ).
−
By Conclusion (5± ), generic leaves of Λ−
ν are contained in U , and generic leaves
+
−
−
of Λ+
and U + are disjoint from Λ±
ν are contained in U . Since U
η , neither Λν
+
−
+
−
+
±
nor Λν is in {Λη , Λη } and so Λν and Λν are geometric. By Conclusion (6), Λν is

34

a dual lamination pair for ν, and so Ana Λ±
ν is a well-defined vertex group system.
±
±
±
±
By Conclusion (1), Ana Λ±
⊏
A
Λ
and
A
na µ
na Λν ⊏ Ana Λβ and (ν, Λν ) ∈ ΘL . The
ν
±
containment Ana Λ±
µ ⊐ Ana Λν is therefore proper, in contradiction to our choice
of µ.
Free subgroups. We will need the following Proposition 4.7 which, inside certain subgroups of Out(Fn ), produces useful free subgroups. In the nongeometric case
this is a consequence of a theorem of Ghosh [Gho16]. The geometric case combines
tools from our subgroup decomposition theory in [HM17f] with a result of Farb and
Mosher [FM02] that produces useful free subgroups of mapping class groups.
Proposition 4.7. Given φ, ψ ∈ Out(Fn ) and a proper free factor system F that is
±
±
±
preserved by φ and ψ, and given lamination pairs Λ±
φ ∈ L (φ) and Λψ ∈ L (φ) that
fill Fn , suppose that the following hold:
(1) φ, ψ are both fully irreducible rel F .
+
−
+
(2) {Λ−
φ , Λφ } ∩ {Λψ , Λψ } = ∅;
±
(3) Either both of the lamination pairs Λ±
φ , Λψ are nongeometric, or the group hφ, ψi
is geometric above F .

Then there exists M ≥ 1 such that for any integers m, n ≥ M the outer automorphisms φm and ψ n freely generate a rank two free subgroup hφm , φn i < Out(Fn ) such
that any nontrivial ξ ∈ hφm , ψ n i is fully irreducible rel F and has a lamination pair
±
±
±
Λ±
ξ that fills rel F , and if both of Λφ , Λψ are nongeometric then Λξ is nongeometric.
Remark. In a more restricted context contained in the proof of Proposition 4.9,
we shall prove a stronger conclusion saying that, after further increasing M, each pair
Λ±
ξ fills Fn in the absolute sense.
Proof. The result breaks naturally into two cases.
±
Case 1: Λ±
φ , Λψ are both nongeometric. The conclusion in this case exactly
matches the conclusion of [Gho16, Theorem 7.3]. Our work here is therefore just to
±
verify the hypothesis of that theorem, which is that (φ, Λ±
φ ) and (ψ, Λψ ) are independent rel F [Gho16, Definition 7.2]. We verify each of the six clauses of independence
rel F .
±
Independence rel F , clause (1): Neither of Λ±
φ , Λψ is carried by F .
±
Independence rel F , clause (2): {Λ±
φ } ∪ {Λψ } fill rel F .
±
These hold because each pair Λ±
φ and Λψ individually fills.

35

Independence rel F , clauses (3,4): Generic leaves of Λ±
φ are weakly attracted to
−
+
−1
Λψ by iteration of ψ and to Λψ by iteration of ψ, and similarly with φ, ψ
switched.
This follows from Fact 4.3 after checking its hypotheses. Hypothesis (1) of Fact 4.3 is
identical to Proposition 4.7 (2). Hypothesis (2) of Fact 4.3 follows from Fact 4.2 (3)
using that a filling generic leaf ℓ of an attracting lamination cannot be carried by
a component of a proper free factor system, nor by the conjugacy class of a cyclic
subgroup because ℓ is nonperiodic.
±
Independence rel F , clause (5): The free factor systems Ana Λ±
φ = Ana Λψ are
mutually malnormal rel F .

The meaning of mutual malnormality of two free factor systems A1 and A2 rel F is
that for each subgroup C < Fn , if [C] ⊏ A1 and [C] ⊏ A2 then [C] ⊏ F . In the
±
current situation is is obvious from the fact that Ana Λ±
φ = Ana Λψ = F (Fact 4.2 (3)).
±
Independence rel F , clause (6): Both lamination pairs Λ±
φ , Λψ are nongeometric.

This holds by assumption of Case 1, and so Case 1 is complete.
±
Case 2: Λ±
φ and Λψ are not both nongeometric. Denote Γ = hφ, ψi <
Out(Fn ). Combining hypothesis (3) of the proposition with the hypothesis of Case 2
it follows that Γ is geometric above F . Also, the group Γ is fully irreducible rel F ,
because it contains an element which is fully irreducible rel F , namely φ. The hypotheses of [HM17f, Theorem J] for the subgroup Γ are therefore satisfied.
From the conclusions of [HM17f, Theorem J] we obtain a compact surface S with
nonempty boundary and an injection π1 S ֒→ Fn whose image is its own normalizer,
such that Γ stabilizes the subgroup system [π1 S]. It follows from [HM17c, Fact 1.4]
that there is a well-defined homomorphism Γ → Out(π1 S) which assigns to each ξ ∈ Γ
the element of Out(π1 S) that is represented by the element of Aut(π1 S) obtained by
choosing an automorphism Ξ ∈ Aut(Fn ) that represents ξ and preserves π1 S, and
restricting Ξ to π1 S. Also, the image of this homomorphism is contained in the natural
MCG(S) subgroup of Out(π1 S) thereby giving a homomorphism dj # : Γ → MCG(S).
Also, dj # (ξ) is pseudo-Anosov if and only if ξ is fully irreducible rel F . Also, the
induced map djB : B(π1 (S)) → B(Fn ) = B induces a bijection between the following
two sets: the set of all geodesic laminations on S which are unstable laminations of
pseudo-Anosov elements of dj # (Γ) (here we pick a hyperbolic structure on S with
totally geodesic boundary); and the set of all attracting laminations not supported
by F of elements of Γ that are fully irreducible rel F .
±
Consider dj # (φ), dj # (ψ) ∈ dj # (Γ) < MCG(S). Their lamination pairs Λ±
φ , Λψ
form four distinct closed subsets of B, and therefore the unstable/stable geodesic
st
un
st
laminations pairs Λun
φ , Λφ , Λψ , Λψ form four distinct laminations on S. The hypotheses of [FM02, Theorem 1.4] are therefore satisfied, the conclusions of which give

36

the existence of M ≥ 1 such that for all m, n ≥ M the mapping classes dj # (φm ),
dj # (ψ n ) freely generate a rank 2 subgroup of dj # (Γ) such that any nontrivial element is pseudo-Anosov. It follows that φm , ψ n freely generate a rank 2 subgroup
of Γ such that any nontrivial element ξ is fully irreducible rel F , and has a unique
attracting/repelling lamination pair Λ±
ξ not supported by F which maps via djB to
st
#
the unique unstable/stable lamination pair Λun
ξ , Λξ of dj (ξ) in S. Since ξ is fully
±
irreducible rel F , this pair Λξ fills rel F . Case 2 is now complete.

4.2

Proof that Theorem E implies Theorem B.

Throughout this section we fix the following notation, taken from the hypotheses of
Theorem B:
(1)Γ Γ < IAn (Z/3) is an infinite lamination group with (virtually) abelian restrictions.
(2)Γ A is a maximal, Γ-invariant, proper free factor system of Fn .
Without further reference we make use of the fact [HM17d, Theorem 3.1] that if
η ∈ IAn (Z/3) then every free factor conjugacy class that is η-periodic is fixed by η.
Since Γ actually has abelian restrictions (Corollary 4.1), we have:
(3)Γ For each component [A] of A,
• [A] is fixed by Γ.
• The group ΓA = Image(Γ 7→ Out(A)) is abelian.
We will need the following minor extension of [FH09, Lemma 4.4].
Lemma 4.8. Each virtually abelian subgroup A of Out(Fn ) is a finite lamination
subgroup. Furthemore if A < IAn (Z/3) then each element of L(A) is A-invariant.
Proof. By [FH09, Corollary 3.14] there is a finite index abelian subgroup A′ < A that
is generated by rotationless elements. Applying [FH09, Lemma 4.4] it follows that
L(A′ ) is a finite collection of A′ -invariant laminations. Since each φ ∈ A′ has a power
φk ∈ A, k 6= 0, and since L(φ) = L(φk ), it follows that L(A′ ) = L(A) and so L(A)
is finite.
Suppose now that A < IAn (Z/3), that Λ ∈ L(A), and that ψ ∈ A; we prove
that ψ(Λ) = Λ. Choose φ ∈ A so that Λ ∈ L(φ) = {Λ1 , . . . , Λm }. Since A is
virtually abelian there exists k ≥ 1 such that ψ k commutes with φk . It follows that
ψ k permutes L(φ). The free factor supports of the Λi ’s are distinct [BFH00, Lemma
3.2.4] and are permuted by ψ k . Since ψ ∈ IAn (Z/3), it preserves the free factor
support of each Λi and so ψ preserves each Λi .

37

Returning now to the context of the subgroup Γ in property (1)Γ above, by
Lemma 4.8 it follows that Γ is not virtually abelian (and hence that hypothesis is not
needed in the statement of Theorem B, as it was in Theorem C).
Applying Lemma 4.8 together with (3)Γ we have:
(4)Γ For each proper free factor A < Fn such that Γ < Stab[A], the image of
the homomorphism Γ ֒→ Stab[A] 7→ Out(A) is a finite lamination group. In
particular, for each component [A] of A the subgroup ΓA < Out(A) is a finite
lamination group.
Given ξ ∈ Γ consider the set L(ξ; A) consisting of those laminations in L(ξ) that are
not carried by A. Let L(Γ; A) = ∪ξ∈Γ L(ξ; A), which is an infinite set, because L(Γ)
is infinite by hypothesis, but by (4)Γ only finitely many elements of L(Γ) are carried
by A. We have:
(5)Γ A ⊏ {Fn } is a multi-edge extension (by Fact 4.2 (3a)), and every element of
L(Γ; A) has infinite orbit under Γ (by Proposition 4.6).
For each Λ ∈ L(Γ; A) consider the free factor system Fsupp (Γ · Λ), that is, the
smallest free factor system carrying every lamination in the Γ-orbit of Λ. Clearly
Fsupp (Γ · Λ) is Γ-invariant. By (1)Γ we have Γ < IAn (Z/3), and so each component
of Fsupp (Γ · Λ) is Γ invariant. By item (5)Γ , one of the finitely many components of
Fsupp (Γ · Λ) supports infinitely many elements of L(Γ; A) and so the restriction of Γ
to that component is not a finite lamination group; by (4)Γ that component must be
{[Fn ]}. This shows:
(6)Γ The Γ-orbit of each element of L(Γ; A) fills Fn .
Using this we next verify most of conclusion (2) of Theorem B:
(7)Γ There exists η ∈ [Γ, Γ] such that η is fully irreducible rel A and such that η
acts loxodromically on F S(Fn ).
We prove (7)Γ in two cases depending on whether Γ is geometric above A (see just
before Theorem 4.5 to recall the definition).
Case 1: Γ is not geometric above A. Applying [HM17f, Proposition 2.2
(1)], we obtain η ∈ Γ and a non-geometric lamination pair Λ±
η ∈ L(η; A) such that
Ana (Λ±
)
=
A.
We
may
assume
in
addition
that
η
is
chosen
so that Fsupp (Λ±
η
η ) is
maximal with respect to ⊏.
We claim that the free factor system Fsupp (Λ±
η ) is Γ-invariant. If not, then there
±
±
+
exists ζ ∈ Γ such that Fsupp (Λη ) 6= ζ(Fsupp(Λη )). Since Fsupp (Λ±
η ) = Fsupp (Λη )) =
−
+
−
+
−
Fsupp (Λη ), and similarly with ζ applied, it follows that {Λη , Λη }∩{ζ(Λη ), ζ(Λη )} = ∅.
Applying [HM17f, Inductive Step of Proposition 2.4 (2)] it follows that if the integer
m > 0 is sufficiently large then η ′ = ζη mζ −1η −m has a nongeometric lamination pair
38

±
±
±
Λ±
η′ such that Ana (Λη′ ) = A and such that Fsupp (Λη ) is strictly contained in Fsupp (Λη′ ),
contradicting maximality and therefore proving the claim.
±
Applying Γ-invariance of Fsupp (Λ±
η ), for each θ ∈ Γ we have Fsupp (θ · Λη ) =
±
±
±
±
θ(Fsupp (Λη )) = Fsupp (Λη ) and so Fsupp (Λη ) = Fsupp (Γ · Λη ) = {[Fn ]} where the latter equation follows from (6)Γ . Thus Λ±
η fills Fn and so η acts loxodromically on
F S(Fn ), by [HM14b].
If η were not fully irreducible rel A, that is if there existed an η-invariant free factor
system A′ contained strictly between A and {[Fn ]} = Fsupp (Λ±
η ), then any conjugacy
′
class carried by A but not by A would not be weakly attracted to Λ±
η , contradicting
Fact 4.2 (2).
Lastly, it remains to arrange that η ∈ [Γ, Γ] (if it is not already true). Applying
−
+
−
(5)Γ we may choose ζ ∈ Γ such that {ζ(Λ+
η ), ζ(Λζ )} ∩ {Λη , Λη } = ∅. Applying
[HM17f, Inductive Step of Proposition 2.4] again, if the integer m > 0 is sufficiently
large then ζη m ζ −1η −m ∈ [Γ, Γ] satisfies all the portions of (7)Γ already established
for η.
Case 2: Γ is geometric above A. The proof is similar to Case 1 but cites
different results from [HM17f]. Applying [HM17f, Proposition 2.2 (b)], there exists η ∈ Γ which is fully irreducible rel A and which has a geometric lamination
±
pair Λ±
η ∈ L(η; A) whose nonattracting subgroup system Ana (Λη ) is Γ-invariant,
that is, Stab(Ana (Λ±
η )) = Γ. Also, the nonattracting subgroup system has the form
Ana (Λ±
)
=
A
∪
{[C]}
for some rank 1 subgroup C < Fn [HM17f, Proposition 2.2
η
(b)(iii)]. Note that [C] 6∈ A, for otherwise it would follow that Ana (Λ±
η ) = A is a free
factor system, contradicting Fact 4.2 (1).
We will need that there is no vertex group system A′ strictly contained between
A and A ∪ {[C]}, for suppose that A ⊏ A′ ⊏ A ∪ {[C]}. Using that vertex group
systems are malnormal [HM17c, Lemma 3.1], it follows that A ⊂ A′ . If A′ = A we
are done. Otherwise, consider any component [C ′ ] ∈ A′ − A. From malnormality it
follows that [C ′ ] ⊏ [C], and so up to conjugacy we may assume that C ′ < C and
hence C, C ′ each have rank 1. By [HM17c, Proposition 3.2] it follows that C ′ = C
and hence A′ = A ∪ {[C]}.
By applying [HM17f, Proposition 2.3 (3)(b)] we conclude that Fsupp (Λ±
η ) is Γinvariant. Using the same argument as in Case 1, it follows that Λ±
fills
F
n , and so
η
η acts loxodromically on F S(Fn ) [HM14b].
−
To arrange that η is in [Γ, Γ], as before choose ζ ∈ Γ such that {ζ(Λ+
η ), ζ(Λζ )} ∩
−
{Λ+
η , Λη } = ∅. Applying [HM17f, Induction Step of Proposition 2.2], if the integer
m > 0 is sufficiently large then η ′ = ζη mζ −1 η −m ∈ [Γ, Γ] has a geometric lamination
′
pair Λ±
η′ ∈ L(η ; A) whose nonattracting subgroup system satisfies the containment
relations
±
A ⊏ Ana (Λ±
η′ ) ⊏ Ana (Λη ) = A ∪ {[C]}

As shown above, one of these containment relations is an equation, and it cannot be
±
the first because Λ±
η′ is geometric and so Ana (Λη′ ) is not a free factor system. Thus
±
±
Ana (Λη′ ) = Ana (Λ±
η ) and so Stab(Ana (Λη′ )) = Γ. Applying [HM17f, Lemma 2.3 (3)]
39

it follows that η ′ is fully irreducible rel A and that Fsupp (Λ±
η′ ) is Γ-invariant, and so
again by the same argument as in Case 1 it follows that η ′ acts loxodromically on
F S(Fn ).
To complete the proof of Theorem B there are just two more properties to verify:
(8)Γ Every η as in (7)Γ — that is, every η ∈ [Γ, Γ] that is loxodromic and fully
irreducible rel A — is a WWPD element for the action Γ y F S(Fn ).
(9)Γ The action Γ y F S(Fn ) is nonelementary.
To prove item (8)Γ , note that for each component [A] of A the restriction of η to
Out(A) is in the commutator subgroup of the abelian group ΓA and hence is trivial.
Applying Theorem E it follows that η is WWPD.
To prove item (9)Γ, take η as in (8)Γ , let Λ±
η be its filling lamination pair, and apply
(5)Γ to conclude that the Γ orbit of the pair Λ±
η is infinite. In particular, for some δ the
±
±
±
pair Λη is disjoint from the pair δ(Λη ) = Λδηδ−1 which also fills. Thus η δ = δηδ −1 is
also loxodromic. Furthermore η, η δ form an independent pair of loxodromics, because
by [HM14b] the set of attracting (repelling) points of loxodromic elements of Γ in the
Gromov boundary ∂ F S(Fn ) corresponds bijectively and Γ-equivariantly to the set of
repelling (attracting) laminations Λ ∈ L(Γ) such that Λ fills Fn .

4.3

Proof that Theorems B, C, D imply Theorem A

Given a finitely generated subgroup G < Out(Fn ) which is not virtually abelian, to
prove Theorem A it suffices to verify the global WWPD hypothesis for the group G
(Definition 2.13), for then we can apply Theorem D from which we obtain an embedding ℓ1 ֒→ Hb2 (G; R). It remains to show how to apply Theorems B and C to verify
the global WWPD hypothesis for G.
Setup: Throughout this section we denote the following objects satisfying various
properties:
• Denote G0 = G ∩ IAn (Z/3), a finite index normal subgroup of G. It follows that
G0 is not abelian.
• Choose Fr < Fn to be a free factor of rank r 6= n, with corresponding restriction
homomorphism π : Stab[Fr ] → Out(Fr ), such that the following two properties hold
and the rank r is minimal with respect to these properties:
— G0 < Stab[Fr ],
— the group Γ = π(G0 ) < Out(Fr ) is not virtually abelian.
By restriction we have a surjective homomorphism π : G0 → Γ. Since G0 is contained
in IAn (Z/3) ∩ Stab[Fr ], and since the subgroup π IAn (Z/3) ∩ Stab[Fr ] < Out(Fr ) is
contained in IAr (Z/3), and since virtually abelian subgroups of IAr (Z/3) are abelian,
[HM17g] we have
40

• Γ < IAr (Z/3).
By choice of the rank r we have:
• Γ has virtually abelian restrictions.
To prove this, if B < Fr is a proper free factor such that Γ < StabOut(Fr ) [B] then
by malnormality of Fr it follows also that G0 < StabOut(Fn ) [B], and by minimality of
r it follows that Image(Γ 7→ Out(B)) = Image(G0 7→ Out(B)) is virtually abelian.
Combining the last two bullet points with Corollary 4.1 we have:
• Γ has abelian restrictions.
With this setup, henceforth we work primarily in Out(Fr ), thinking of G and its subgroups somewhat abstractly rather than as subgroups of Out(Fn ). In particular, our
strategy for verifying the global WWPD hypothesis of G is to work with a hyperbolic action of (a certain subgroup of) G0 that factors through a hyperbolic action
of (a certain subgroup of) Γ < Out(Fr ), the latter action being obtained by applying
Theorem B or C. As a notational side effect, many standard notations should be
interpreted in Fr , for example the attracting lamination notation L(·).
The proof now breaks into two cases, depending on whether Γ is an infinite lamination subgroup of Out(Fr ).
Case 1: Γ < Out(Fr ) is a finite lamination subgroup. Applying Theorem C,
we obtain the following objects: a nontrivial free factor A < Fr with Γ-invariant
b = ρ(Γ) < Aut(A), a
conjugacy class, a homomorphism ρ : Γ 7→ Aut(A) with image H
b and an action N ′ y X on a hyperbolic space. Also
finite index subgroup N ′ < H,
from Theorem C we have the following:
(a)′ Every element of N ′ acts elliptically or loxodromically on X;

(b)′ The action N ′ y X is nonelementary;
(c)′ Every element of [N ′ , N ′ ] is either elliptic or WWPD with respect to the action
N ′ y X.
Let N < G0 < G be a finite index normal subgroup of G such that ρ ◦ π(N) < N ′ ; for
instance, we can take N to be the intersection of all the G-conjugates of (ρ◦ π)−1(N ′ ).
ρ◦π
By composing the homomorphism N −−→ N ′ with the action N ′ y X we obtain
an action N y X, and clearly properties (a)′ —(c)′ for N ′ imply the same properties
for N:
(a) every element of N acts elliptically or loxodromically on X;
(b) the action N y X is nonelementary;
(c) every element of [N, N] is either elliptic or WWPD with respect to the action
N y X.
41

We shall produce a certain rank 2 subgroup E < [N, N] and for this subgroup we
shall verify Definition 2.13 items (1), (2) and (3)′ , which proves the global WWPD
hypothesis for G. Definition 2.13 (1) automatically holds because it is the same as
property (a).
Let i1 , . . . , iK ∈ Aut(N) be outer representatives of the inner action of G on N,
iκ
with i1 = IdN . Let N yκ X denote the composed action N −
→
N y X; so N y1 X
is another notation for the given action N y X. We also use yκ to denote restriction
of the action N yκ X to subgroups of N. Since [N, N] is a characteristic subgroup
of N, it follows that item (c) above holds not just for the given action N y1 X but
for each of the composed actions N yκ X. That is,
(d) For every a ∈ [N, N] and every κ = 1, . . . , K, the element a is either elliptic or
WWPD with respect to the action N yκ X.
Apply (b) to obtain an independent pair of loxodromic elements in N, and then apply
hyperbolic ping pong to this pair of elements to obtain a rank 2 Schottky subgroup
E0 < N. The commutator subgroup [E0 , E0 ] < E0 is free of infinite rank and hence
the restricted action [E0 , E0 ] is nonelementary. Since [E0 , E0 ] < [N, N], we may pick
an independent pair of loxodromic elements in [N, N], and then apply hyperbolic ping
pong to obtain a rank 2 Schottky subgroup E1 < [N, N]. Every nonidentity element
of E1 is loxodromic, and by applying (c), every nonidentity element of E1 satisfies
WWPD, thus every rank 2 subgroup E ⊂ E1 satisfies Definition 2.13 (2).
To complete the verification of the WWPD hypothesis it remains to describe a
rank 2 subgroup E < E1 that satisfies Definition 2.13 (3)′ , and by applying (d) it
suffices to show that E satisfies the following:
• Each restricted action E yκ X is either Schottky or elliptic, for κ = 1, . . . , K.
We do this with an argument of Bestvina and Fujiwara extracted from the proof of
[BF02, Theorem 8], as follows. Assume by induction that for some κ = 1, . . . , K − 1
we have a rank 2 subgroup Eκ ⊂ E1 such that for each i = 1, . . . , κ the action
Eκ yi X is either Schottky or elliptic, and hence the restriction of the action yi
to any rank 2 subgroup of Eκ is either Schottky or elliptic. We break into cases
depending on the nature of the action Eκ yκ+1 X. If that action is elliptic then,
taking Eκ+1 = Eκ , the induction is complete. If that action is nonelementary then,
picking independent loxodromic elements and using hyperbolic ping-pong, we obtain
a rank 2 subgroup Eκ+1 < Eκ for which the action Eκ+1 yκ+1 X is Schottky, and
the induction is complete. In the remaining case the action Eκ yκ+1 X is nonelliptic
and elementary, and so there exists a point ξ ∈ ∂X which is either an attractor or
repeller for every loxodromic element of the action Eκ yκ+1 X. By combining the
inclusion Eκ ⊂ [N, N] with property (d) and Proposition 2.6 (5), it follows that there
exists η 6= ξ ∈ ∂X such that {ξ, η} is the attractor repeller pair for every loxodromic
element in Eκ . In this situation the translation number homomorphism τ : Eκ → R
42

is defined by picking a base point x ∈ X, and for each θ ∈ Eκ setting

n

if ξ is the attractor for θ
 limn→∞ d(x, θ (x))/n
τ (θ) = − limn→∞ d(x, θn (x))/n
if η is the attractor for θ


0
otherwise

Since Image(τ ) is abelian, Ker(τ ) is an infinite rank subgroup of Eκ . Taking Eκ+1 <
Ker(τ ) to be any rank 2 subgroup, the action Eκ+1 yκ+1 X is elliptic and the
induction is complete.
Case 2: Γ < Out(Fr ) is an infinite lamination subgroup. Letting A be any
maximal, proper, Γ-invariant free factor system in Fr , the hypotheses of Theorem B
are satisfied, and from its conclusions we obtain the following:
(a) The action Γ y F S(Fr ) is nonelementary;
(b) There exists a loxodromic element of [Γ, Γ] which is fully irreducible rel A, and
any such element satisfies WWPD for the action Γ y F S(Fr )
We may apply all of the numbered properties (1)Γ – (9)Γ from Section 4.2, each of
which was proved starting only with the assumption that Γ and A satisfy the hypotheses of Theorem B.
We also adopt all the notation from the setup at the beginning of Section 4.3, but
to simplify notation and highlight parallels with Case 1 we use the notation
N = G0
π

We have an action N y F S(Fr ), given by the composition N −
→ Γ y F S(Fr ).
Evidently the following properties corresponding to (a) and (b) are satisfied:
(a)N The action N y F S(Fr ) is nonelementary;
(b)N There exists a loxodromic φ ∈ [N, N] such that π(φ) ∈ Out(Fr ) is fully irreducible rel A, and any such φ satisfies WWPD for the action N y F S(Fr ).
The proof in Case 2 will parallel Case 1 for a brief while, before diverging. Choose
i1 , . . . , iK ∈ Aut(N) to be outer representatives of the inner action of G on N, with
iκ
i1 = IdN . Let N yκ F S(Fr ) denote the composed action N −
→
N y F S(Fr ), and
use the same action symbol yκ for restrictions of N yκ F S(Fr ) to subgroups of N.
Definition 2.13 (1) is simply property (a)N above. Thus to verify the global WWPD
hypothesis for G we must produce a rank 2 subgroup E < [N, N] and use it to verify
the following parts of Definition 2.13:
(2) The restricted action E y F S(Fr ) is Schottky and its nontrivial elements all
satisfy WWPD with respect to the action N y F S(Fr ).
43

(3)′ For each κ = 1, . . . , K, the action E yκ F S(Fr ) is either elliptic, or it is
Schottky and its nontrivial elements all satisfy WWPD with respect to the
action N yκ F S(Fr ).
Where Cases 1 and 2 diverge is that we do not know that every loxodromic φ ∈ [N, N]
is a WWPD element for the action N y F S(Fr ); we know this only for those
φ such that π(φ) ∈ Out(Fr ) is fully irreducible rel A. Thus, to verify the global
WWPD hypothesis for G using the action N y F S(Fr ) there is still quite a bit of
intricate work to do involving subgroup decomposition theory, lamination ping-pong,
and Ghosh’s theorem.
iκ
π
Define πκ : N → Γ to be the composition N −
→
N−
→ Γ. The action N yκ F S(Fr )
πκ
is thus the same as the composed action N −→
Γ y F S(Fr ). For each component
[A] of A, since the restriction of Γ to Out(A) is abelian, it follows that the restriction
of [Γ, Γ] to Out(A) is trivial. Since the characteristic subgroup [N, N] is preserved by
the isomorphism iκ , we have πκ [N, N] = π[N, N] < [Γ, Γ] for κ = 1, . . . , K, and so
(c)N For each component [A] of A and each κ = 1, . . . , K, the restriction of πκ [N, N]
to Out(A) is trivial.
Consider a ∈ N, and denote its images in Γ as ακ = πκ (a) ∈ Γ, 1 ≤ κ ≤ K.
We say that κ is a PG index for a if L(ακ ; A) = ∅; otherwise κ is an EG index for a.
Assuming that κ is an EG index for a, we say that κ is a nongeometric index for a if
some element of L(ακ ; A) is non-geometric; otherwise κ is a geometric index for a. Let
t1 ≥ 0 be the maximum number of non-geometric indices that occurs for any element
of N, and let M′ be the set of all a ∈ N having t1 non-geometric indices; note that
t1 = 0 if and only if the subgroup Γ < Out(Fr ) is geometric above A. Let t2 ≥ t1 be
the maximum number of EG indices that occur for some a ∈ M′ , and let M be the
set of all a ∈ M′ having t2 EG indices. There exists a ∈ N and κ ∈ {1, . . . , K} such
that κ is an EG index for a, because by property (a) some element of Γ is loxodromic
and so has a filling lamination. The set M is therefore nonempty and t2 ≥ 1. After
permuting the κ’s if necessary we may assume that the following subset of M is
nonempty:
M0 = {a ∈ N

κ is a non-geometric index of a for 1 ≤ κ ≤ t1 , and
κ is a geometric index of a for t1 < κ ≤ t2 }

That is, a ∈ M0 if and only if: L(ακ ; A) has a nongeometric lamination for each
1 ≤ κ ≤ t1 ; L(ακ ; A) is a nonempty set of geometric laminations for t1 < κ ≤ t2 ; and
L(ακ ; A) is empty for t2 < κ ≤ K.
Given a ∈ M0 with ακ = πκ (a), an assignment of lamination pairs for a is a
function
κ 7→ Λ±
ακ ∈ L(ακ ; A) defined for 1 ≤ κ ≤ t2 ,


±
denoted in shorthand as Λ±
ακ . We say that this assignment Λακ is M0 -consistent
if Λ±
ακ is non-geometric for each 1 ≤ κ ≤ t1 .
44

Proposition 4.9.

(1) There exists a ∈ M0 with an M0 -consistent assignment Λ±
ακ of filling lamination pairs, such that ακ = πκ (a) ∈ Γ is fully irreducible rel A for each
1 ≤ κ ≤ t2 .
(2) Either t1 = 0 or t2 = t1 ≥ 1. More precisely: either each Λ±
ακ is non-geometric
±
for each κ; or Γ is geometric above A and therefore Λακ is geometric for each κ.
Before proving Proposition 4.9, we apply it to the construction of a rank 2 subgroup
E < [N, N] that satisfies (2) and (3)′ .

Choose a ∈ M0 and an assignment of lamination pairs Λ±
ακ satisfying Proposition 4.9 (1). Consider the set Cr of closed subsets of B(Fr ), on which Out(Fr ) acts
+
naturally. Each lamination Λ−
ακ , Λακ for 1 ≤ κ ≤ t2 is an element of Cr . By [HM14b,
+
Corollary 1.3] we have equality of stabilizer subgroups StabΓ (Λ−
ακ ) = StabΓ (Λακ ), and
−
by property (5)Γ this subgroup has infinite index in Γ. The subgroup StabΓ {Λακ , Λ+
ακ },
+
−
which contains StabΓ (Λακ ) = StabΓ (Λακ ) with index at most 2, therefore also has infinite index in Γ.
πκ
For each 1 ≤ κ ≤ t2 , consider the composed action N −→
Γ < Out(Fr ) y Cr ,
denoted as N yκ Cr . The Second Sublemma of Lemma 2.1 of [HM17f] applied to
the actions N yκ Cr produces an element b ∈ N such that for all 1 ≤ κ ≤ t2 , letting
βκ := πκ (b), we have
−
+
−
βκ ({Λ+
ακ , Λακ }) 6= {Λακ , Λακ }

and therefore by the Independence Theorem [HM14b, Theorem 1.2] we have
(∗)

−
+
−
βκ ({Λ+
ακ , Λακ }) ∩ {Λακ , Λακ } = ∅

±
±
Let c = bab−1 ∈ N, let γκ = πκ (c) = βκ ακ βκ−1 , and let Λ±
ψκ = βκ (Λακ ) ∈ L (γκ ; A).
+
−
+
−
Then γκ is fully irreducible rel A and {Λακ , Λακ } ∩ {Λγκ , Λγκ } = ∅ for all 1 ≤ κ ≤ t2 ;

moreover, Λ±
γκ is an M0 -consistent assignment of filling lamination pairs for c.
We now apply Proposition 4.7 to the pair of elements ακ and γκ for each 1 ≤
κ ≤ t2 , noting that we have already verified hypotheses (1) and (2) of Proposition 4.7
for that pair, and that hypothesis
(3) follows from M0 -consistency of each of the

±
±
assignments Λακ and Λγγ and from Proposition 4.9 (2). From the conclusion of
Proposition 4.7, for each 1 ≤ κ ≤ t2 there exists Mκ so that for each m ≥ Mκ the
subgroup Eκ′ of Γ generated by ακm and γκm is free of rank 2, each non-trivial element
ξκ ∈ Eκ′ is irreducible rel A, each ξκ has a unique lamination pair Λ±
ξκ that fills rel
±
A, and Λξκ is nongeometric for 1 ≤ κ ≤ t1 and geometric for t1 < κ ≤ t2 . Increasing
each Mκ to M = maxκ Mκ , we obtain a subgroup E ′ of N freely generated by am
′
′
and cm such that that Eκ′ = πκ (E
 ), and such that for each nontrivial x ∈ E , letting
±
ξκ = πκ (x), the assignment Λξκ is M0 -consistent.
By [HM14b, Theorems 1.1 and 1.2], the actions of ακ and γκ on F S(Fr ) are loxodromic and independent. By hyperbolic ping-pong (see the beginning of Section 2.4),

45

after a further increase of M, we may also assume that the action of each Eκ′ on
F S(Fr ) is Schottky. In particular, each non-trivial ξκ ∈ Eκ′ has a filling lamination
pair, and that pair must be Λ±
ξκ since all other lamination pairs for ξκ are supported
by A. It follows that for each 1 ≤ κ ≤ t2 the restricted action E ′ yκ F S(Fr ) is
Schottky and each non-trivial element of πκ (E ′ ) is irreducible rel A. Moreover each
non-trivial element of E ′ is contained in M0 .
Choose a rank 2 free subgroup E < [E ′ ∩ N, E ′ ∩ N] < [N, N]. Note that each
restricted action E yκ F S(Fr ) is still Schottky for each 1 ≤ κ ≤ t2 , and each nontrivial element of E is contained in M0 . Applying property (c)N , for each component
[A] of A the restriction of each πκ (E) to Out(A) is trivial. Theorem E therefore
implies that E yκ F S(Fr ) is a WWPD Schottky group for each 1 ≤ κ ≤ t2 . Since
each non-trivial element of E is contained in M0 , the actions E yκ F S(Fr ) must
be elliptic for κ > t2 . This completes the verification of (2) and (3)′ using E, thus
finishing Case 2 subject to the proof of Proposition 4.9.
Proof of Proposition 4.9. We start with two lemmas needed to set up the pingpong proof. Most of the following lemma is cited from [HM17f, Lemma 2.3 (3)].
Lemma 4.10. Suppose that Γ < IAr (Z/3) is irreducible relative to a free factor
system A, and suppose that φ ∈ Γ is rotationless and has a geometric lamination pair
±
Λ±
φ ∈ L(φ; A) such that Ana (Λφ ) is Γ-invariant. Then
(1) φ is irreducible rel A [HM17f, Lemma 2.3 (3)(a)].
(2) The free factor support of Λ±
φ is Γ-invariant [HM17f, Lemma 2.3 (3)(b)].
(3) Ana Λ±
φ = A∪{[C]} where C is a maximal infinite cyclic subgroup of Fr [HM17f,
Lemma 2.3 (3)(c)].
(4) Γ is geometric above A.
Proof. To prove (4), if Γ is not geometric above A then by [HM17f, Proposition 2.4]
there exists a rotationless θ ∈ Γ that is irreducible rel A and there exists Λ±
θ ∈ L(θ; A)
±
such that Ana (Λθ ) = A. But this contradicts the fact that [C] is θ-invariant and not
carried by the free factor system A.
Ping-pong arguments in groups can start with one player — that is, one group element — producing a second player by conjugating the first. The conjugating element
is chosen carefully, depending on the desired outcome. The proof of Proposition 4.9
is a ping-pong game in the group N, carried out inductively: given ai ∈ M0 , after
choosing a conjugator bi ∈ N we produce the second player ci = bi ai b−1
∈ M0 and
i
m n
then define ai+1 = ci ai for sufficiently large m, n. We then iterate this until the resulting ai satisfies the conclusions of Proposition 4.9. Each step of this iteration can
also be viewed as t2 simultaneous ping-pong games in the quotient group Γ = π(N),
46

with t2 first players ακ (= πκ (a1 )), requiring a very careful and consistent choice of
t2 conjugating elements βκ (= πκ (b1 )). The following lemma describes the choice of
conjugating element b1 . It is a straightforward generalization of [HM17f, Lemma 2.1],
which is used to choose the conjugating maps for the proof of [HM17f, Theorem I].
Lemma 4.11. Suppose that a ∈ M0 and ακ = πκ (a) ∈ Γ, 1 ≤ κ ≤ K. Let
Λ±
ακ ∈ L(ακ ; A) be an M0 -consistent assignment of lamination pairs, with generic
leaves ℓ±
ακ , respectively. There exists b ∈ N such that βκ = πκ (b) satisfies the following
properties for all 1 ≤ κ ≤ t2 .
±
−1 −
−1 +
−
(1) None of the lines βκ (ℓ+
ακ ), βκ (ℓακ ), βκ (ℓακ ), βκ (ℓακ ) is carried by Ana Λακ .
−
+
−
(2) βκ {Λ+
ακ , Λακ } ∩ {Λακ , Λακ } = ∅.
±
±
(3) If Ana Λ±
ακ is not Γ-invariant then βκ (Ana Λακ ) 6= Ana Λακ .
±
±
(4) If FsuppΛ±
ακ is not Γ-invariant then βκ (Fsupp Λακ ) 6= Fsupp Λακ .

Proof. Following the proof of [HM17f, Lemma 2.1], we shall show the following for
all 1 ≤ κ ≤ t2 :
(a)κ There is a finite index subgroup Γκ < Γ such that for any β ∈ Γκ none of the
−
−1 +
±
lines β(ℓ+
(ℓακ ), β −1 (ℓ−
ακ ), β(ℓακ ), β
ακ ), is carried by Ana Λακ .
−
(b)κ StabΓ {Λ+
ακ , Λακ } has infinite index.
±
(c)κ If Ana Λ±
ακ is not Γ-invariant then StabΓ (Ana Λακ ) has infinite index.
±
(d)κ If Fsupp Λ±
ακ is not Γ-invariant then StabΓ (Fsupp Λακ ) has infinite index.

Item (a)κ follows from the First Sublemma in the proof of [HM17f, Lemma 2.1]. Item
(b)κ follows from property (5)Γ . Item (c)κ follows from [HM17f, Lemma 2.3]. Item
(d)κ follows because if StabΓ (Fsupp Λ±
ακ ) has finite index then for each θ ∈ Γ some power
of θ fixes the free factor system Fsupp Λ±
ακ , but since θ ∈ IAn (Z/3) it follows that θ
itself fixes Fsupp Λ±
,
by
[HM17d,
Theorem
3.1].
ακ
By pulling back the infinite index subgroups in (b)κ , (c)κ and (d)κ under the
homomorphism πκ : N → Γ, and letting κ = 1, . . . , t2 vary, we obtain a finite collection
of infinite index subgroups of N. Applying the Second Sublemma in the proof of
[HM17f, Lemma 2.1], there is an infinite subset of N any two elements of which lie in
distinct left cosets of each subgroup in this collection. By the pigeonhole principle,
this subset must contain two elements b1 6= b2 lying in the same left coset of the finite
index subgroup
t2
\
jκ−1 (Γκ )
κ=1

The element b =

b−1
1 b2

satisfies the conclusions of the lemma.
47

Proof of Proposition 4.9. The construction is inductive, using the fact that M0 6= ∅
to choose an arbitrary a(1) ∈ M0 and an M0 -consistent assignment of lamination
pairs Λ±
a sequence a(2), a(3), . . . ∈ M0 with M0 -consistent
πκ (a1 ) , and producing

±
assignments Λπκ (a(i)) . The first part of the proof is a description of a single step
in the induction. The remainder of the proof explains how the induction eventually
terminates in an element of M0 satisfying conclusions (1) and (2) of the proposition.
Assuming that we have produced ai ∈ M
 0 , we describe how to produce ai+1 .
±
Let αi,κ := πκ (ai ) for 1 ≤ κ ≤ t2 . Let Λαi,κ be the corresponding M0 -consistent
assignment of lamination pairs. For notational
 convenience we suppress the subscript
±
“i” for now, writing a, ακ = πκ (a), and Λακ .
Choose b ∈ N as in Lemma 4.11, the conclusions of which we shall apply below.
±
Let βκ = πκ (b), let c = bab−1 , let γκ = πκ (c) = βκ ακ βκ−1, and let Λ±
γκ = βκ (Λακ ) ∈
L(γκ ; A). Fixing m, n ≥ 1 subject to lower bounds to be given, consider x = cm an
and for each 1 ≤ κ ≤ t2 consider ξκ = πκ (x) = γκm ακn .
We want to simultaneously apply Proposition 4.4 (aka [HM17f, Proposition 1.7])
±
to ακ , Λ±
ακ , γκ and Λγκ for 1 ≤ κ ≤ t2 , and so we must check its hypotheses. After
replacing a, and hence c, with an iterate of itself, we may assume for 1 ≤ κ ≤ t2
±
±
±
±
that Λ±
ακ and Λγκ have generic leaves ℓακ and ℓγκ = βκ (ℓακ ) that are fixed by φ and
ψ respectively, with fixed orientation. Note that if 1 ≤ κ ≤ t1 then the pair Λ±
γκ is
±
nongeometric because it is the βκ image of the nongeometric pair Λακ . It remains to
check hypotheses (a) and (i)–(iv) of Proposition 4.4.
±
Hypothesis (a) of Proposition 4.4 says that A ⊏ Ana Λ±
ακ and A ⊏ Ana Λγκ =
±
±
βκ Ana Λ±
ακ . This follows from the assumption that Λακ , and hence also Λγκ , is not
carried by A: every iterate under α and γ of every conjugacy class carried by A is
also carried by A, and so such iterates cannot weakly converge to a lamination that
is not carried by A.
Hypotheses (i)–(iv) of Proposition 4.4 follow from the conclusions of Fact 4.3,
whose two hypotheses we verify by applying the conclusions of Lemma 4.11. Hypothesis (1) of Fact 4.3 is just Lemma 4.11 (2). Hypothesis (2) of Fact 4.3 requires that:
±
±
neither of the lines ℓ±
γκ = βκ (ℓακ ) should be carried by Ana Λακ ; also neither of the
±
±
lines ℓ±
ακ should be carried by Ana Λγκ = βκ Ana Λακ ) which is equivalent to saying
−1 ±
that neither of the lines βκ (ℓακ ) is carried by Ana Λ±
ακ ; but these altogether follow
from Lemma 4.11 (1).
We may now apply the conclusions of Proposition 4.4 for each 1 ≤ κ ≤ t2 ,
producing threshold constants Mκ1 , and if m, n ≥ Mκ1 producing laminations Λ+
ξκ ∈
−
−1
L(ξκ ; A) and Λξκ ∈ L(ξκ ; A).
Define a simultaneous threshold constant M 1 = max1≤κ≤t2 Mκ1 . Fix m, n ≥ M 1 ,
±
and so for each 1 ≤ κ ≤ t2 we have a lamination pair Λ±
ξκ ∈ L (ξκ ).
±
In the case that 1 ≤ κ ≤ t1 , since Λ±
ακ and Λγκ are both nongeometric, Proposi−
tion 4.4 lets us conclude that Λ+
ξκ and Λξκ are both nongeometric. In the case that
+
−
t1 < κ ≤ t2 , the laminations Λξκ and Λξκ are both geometric by definition of M0 . In
either case Proposition 4.4 (6) lets us conclude that Λ±
ξκ form a dual lamination pair.
48


Thus we have shown that Λ±
ξκ is an M0 -consistent assignment of lamination pairs
for x.
Next we show for each 1 ≤ κ ≤ t2 that
(∗) Ana Λξκ ⊏ Ana Λακ . Furthermore, this containment is strict if Ana Λακ is not Γinvariant.
The containment Ana Λξκ ⊏ Ana Λακ follows from Proposition 4.4 (1) which moreover
gives the containment Ana Λξκ ⊏ Ana Λγκ . For the “furthermore” part, if Ana Λακ is
not H-invariant then, by Lemma 4.11 (3), we have
±
±
Ana Λ±
γκ = βκ (Ana Λακ ) 6= Ana Λακ ,

Arguing by contradiction, if furthermore Ana Λξκ = Ana Λακ then we have



Ana Λακ = βκ−1 Ana Λγκ ⊐ βκ−1 Ana Λξκ = βκ−1 Ana Λακ

which must be a strict containment, and by iteration we have an infinite sequence of
strict containments


Ana Λακ ⊐ βκ−1 Ana Λακ ⊐ βκ−1 Ana Λακ ⊐ · · ·

But this contradicts [HM17c, Proposition 3.2] which says any decreasing sequence of
vertex groups systems is eventually constant.
n
Restoring the “i” subscript, and so a = ai and c = ci , now define ai+1 = x = cm
i ai ,
±
±
so ξκ = πκ (ai+1 ) and Λαi+1,κ = Λξκ . This completes the inductive construction of

ai+1 and its M0 -consistent assignment of lamination pairs Λ±
αi+1,κ . We have a
±
±
containment Ana Λ±
αi,κ ⊐ Ana Λi+1,ακ which is strict if Ana Λαi+1,κ is not Γ-invariant.
Consider the whole sequence of containments
±
±
Ana Λ±
α1,κ ⊐ Ana Λα2,κ ⊐ Ana Λα3,κ ⊐ · · ·

Another application of [HM17c, Proposition 3.2] shows this sequence to be eventually
constant, and again applying (∗) it follows that Ana Λ±
αi,κ is eventually Γ-invariant. It
follows that there exists I ≥ 0 independent of κ such that for i ≥ I each Ana Λ±
αi,κ is
Γ-invariant and is independent of i.
We now complete the proof of Proposition 4.9.
Conclusion (2) of Proposition 4.9 follows by observing that if t2 > t1 then, by
Lemma 4.10 combined with Γ-invariance of Ana Λ±
αi,κ for each i ≥ I and each t1 < κ ≤
t2 , the entire group Γ is geometric above A, from which it follows that t1 = 0.
Conclusion (1) will be proved in two cases.
Case 1: t2 > t1 = 0. As just shown, in this case Γ is geometric above A and
Ana Λ±
αi,κ is Γ-invariant for all i ≥ I and all 1 ≤ κ ≤ t2 . Applying Lemma 4.10, it
then follows that αi,κ is irreducible rel A and Λ±
αi,κ has Γ-invariant free factor support
49

and so is filling by property (6)Γ. This proves (1) and so completes the proof of
Proposition 4.9 in Case 1.
Case 2: t2 = t1 . In this case for each 1 ≤ κ ≤ t2 we know that Λ±
αi,κ is non±
geometric for all i, and hence Ana Λαi,κ is a proper free factor system. Combining this
±
with the fact that A ⊏ Ana Λ±
αi,κ , that Γ is irreducible rel A, and that Ana Λαi,κ is
±
Γ-invariant for i ≥ I, it follows that Ana Λαi,κ = A for i ≥ I.
Considering now each 1 ≤ κ ≤ t2 separately, after imposing stricter threshold
constants we wish to prove by induction on j ≥ I that:
±
±
(#) We have containments Fsupp (Λ±
αI,κ ) ⊏ Fsupp (ΛαI+1,κ ) ⊏ · · · ⊏ Fsupp (Λαj,κ )
±
(##) For I ≤ i < j the containment Fsupp (Λ±
αi,κ ) ⊏ Fsupp (Λαi+1,κ ) is proper if and only
if Fsupp (Λ±
αi,κ ) is not Γ-invariant.

Assuming that this has been proved for a particular j, and returning temporarily to the earlier notation where the subscript j has been dropped, we shall apply
the [HM17f, Inductive Step of Proposition 2.4], the hypotheses of which are already known to be true: the pair Λ±
ακ is nongeometric and its nonattracting sub+
−
+
group system equals A; by Lemma 4.11 (2) we have {Λ−
ακ , Λακ } ∩ {Λγκ , Λγκ } =
±
+
−
+
−
{Λακ , Λακ } ∩ {βκ (Λακ ), βκ (Λακ )} = ∅; and there are generic leaves of Λακ that are
fixed by ακ with fixed orientations. From the conclusions of [HM17f, Inductive Step
of Proposition 2.4] we obtain a threshold constant Mκ2 such that if m, n ≥ Mκ2 then
Fsupp (Λακ ) ⊏ Fsupp (Λξκ ) with proper inclusion if and only if Fsupp (Λακ ) is not Γ-invariant.
Letting M 2 = max1≤κ≤t2 Mκ2 , and requiring that m, n ≥ M 2 , for all 1 ≤ κ ≤ t2 we
have completed the inductive verification of (#) and (##) for all j ≥ I.
Since the length of a proper chain of free factor systems of Fr is uniformly bounded,
there exists j ≥ I such that (#) and (##) are satisfied and such that the free factor
±
support of Λ±
αj,κ is Γ-invariant, and so by (6)Γ the lamination pair Λαj,κ fills Fr . Since
Ana Λ±
αj,κ , it follows that αj,κ is irreducible rel A. This being true for each 1 ≤ κ ≤ t2 ,
we have proved conclusion (1) of Proposition 4.9 in Case 2, and so the proof of the
proposition is complete.

5

Well functions and weak tiling functions

In this section and the next we are concerned with the proof of Theorem E. For
this section we assume that n ≥ 3 and we shall pursue a further study of the “well
functions” that were defined in [HM14b, Section 4.4] in the setting of outer automorphisms φ possessing a filling lamination pair Λ±
φ , and that generalize the well
functions defined in [AK11] in the setting of fully irreducible outer automorphisms.
The idea behind a well function is that for any conjugacy class c that is weakly
±1
w(c)
attracted to Λ±
with coarsely wellφ by iteration of φ , there is an iterate φ
w(c)
defined exponent w(c) such that the conjugacy class φ
is not a good approximation
50

+
of either Λ−
φ or Λφ . This contrasts with being outside of the well: for exponents
l >> w(c) the class φl (c) is a good approximation of Λ+
φ but a bad approximation of
−
l
Λφ ; whereas for exponents l << w(c) the class φ (c) is a good approximation of Λ−
φ
+
but a bad approximation of Λφ . The vicinity of the integer w(c) within the whole of
−
the integers Z therefore defines a “well” in which approximations to Λ+
φ and Λφ are
simultaneously at their worst.
Our main result in this section is Proposition 5.7, which allows one to coarsely
evaluate the well function w(c) using a CT representative f : G → G of φ, by studying
how the circuit in G representing c can be “weakly tiled” using natural collections of
paths in G (see Definition 5.4). As an application we prove Corollaries 5.10 and 5.11
which describe strong combinatorial regularity properties of weak tilings. These regularity properties of weak tilings are what we shall use to verify WWPD in Section 6,
in lieu of measure theoretic regularity properties of currents used to verify WPD in
the arguments of [BF02] and [BF10].

5.1

Well functions on intermediate conjugacy classes

We review well functions and their properties from [HM14b, Lemma 4.14], although
see Remark 5.3 to understand some differences of formulation.
Definition 5.1. Given a free factor system F and a conjugacy class c of Fn , we
say that c is filling relative to F if there is no proper free factor system that carries
both c and F . If c is not carried by F nor filling relative to F then c is said to be
intermediate (relative to F ).
We note some properties relating this definition to an outer automorphism φ ∈
Out(Fn ) that preserves F , the first of which is obvious:
• A conjugacy class c is intermediate rel F if and only if φ(c) is intermediate
rel F .
• If φ is fully irreducible rel F and if Λ± is a lamination pair for φ that is not
carried by F and hence fills rel F , then a conjugacy class c is intermediate rel F
if and only if c is weakly attracted to Λ± and is not filling relative to F .
To see why the second item is true, recall [HM17e, Theorem F] which says that c is
weakly attracted to Λ+ under iteration by φ if and only if c is weakly attracted to Λ−
under iteration by φ−1 if and only if c is not carried by the nonattracting subgroup
system Ana (Λ± ). If Λ± is non-geometric then F = Ana (Λ± ) by [HM17e, Remark: The
case of a top stratum] and so “not carried by F ” is equivalent to “weakly attracted
to Λ± ”. If Λ± is geometric then there are conjugacy classes that are not carried by
F and are not weakly attracted to Λ± but each of these fills relative to F ; each such
conjugacy class is represented by a multiple of the peripheral curve in a geometric
model ([HM17c, Proposition 2.18]).
51

For most of this section we focus on the following objects whose notations we fix:
Notation A:
• F is a proper free factor system in Fn , possibly trivial.
• φ, φ−1 ∈ Out(Fn ) are rotationless, and they fix F and are irreducible relative to F .
• Λ± ∈ L± (φ) is a lamination pair that fills Fn .
With this notation we also have the following property:
• F has co-edge number ≥ 2 (Fact 4.2 (3a)).
Definition 5.2. Consider the space of lines B with its weak topology. For any
attracting neighborhood U ⊂ B of a generic leaf of Λ+ , and any conjugacy class c
that is weakly attracted to Λ+ under iteration by φ, there exists a maximal integer
wφ,U (c) such that c ∈ φwφ,U (c) (U) or equivalently φ−wφ,U (c) (c) ∈ U. We refer to wφ,U
as the well function of φ with respect to U. It is immediate from the definition that
wφ,U (φm (c)) = wφ,U (c) + m
for all m.
The two key properties of well functions are as follows:
Coarse well-definedness of wells: For any two U, V ⊂ B attracting neighborhoods of a generic leaf of Λ+ there exists K ≥ 0 so that φK
# (V ) ⊂ U and
K
φ# (U) ⊂ V . It follows that
|wφ,U (c) − wφ,V (c)| ≤ K
for all c that are weakly attracted to Λ± . In other words the function wφ,U (·) is
coarsely well-defined independent of U.
Coarse additive inverse property of wells: For any attracting neighborhoods U ±
of generic leaves of Λ± , respectively, there exists L ≥ 0 so that for all intermediate conjugacy classes c we have
|wφ,U + (c) + wφ−1 ,U − (c)| ≤ L
In other words, the well functions wφ,U + (·) and wφ−1 ,U − (·) are coarse additive
inverses of each other.
The coarse well-definedness property is obvious. The coarse additive inverse property
is proved in [HM14b, Lemma 4.14(1)] (which is in turn based on [HM17f, Proposition
3.1]) for a particular choice of U ± , and so it holds for all choices. Note that coarse
well-definedness does not require the hypothesis of “intermediate”; that hypothesis
need only be brought in for results, like the coarse additive inverse property, that
ultimately depend on [HM17f, Proposition 3.1].
52

Henceforth, when attracting neighborhoods of generic leaves of Λ+ and of Λ− are
chosen, implicitly or explicitly, we write the corresponding well functions of φ and φ−1
as wφ (·) and wφ−1 (·), suppressing the dependence on the choice of attracting neighborhoods.
Remark 5.3. Our presentation here of well functions differs from the presentation in
[HM14b] version 1 in a few regards. One difference is that well functions were defined
there only for particular choices of U − , U + ; the coarse well-definedness property lets
us use any choices. Another difference is that wφ as defined here equals the additive
inverse of wφ−1 as it was defined in Lemma 4.14 of [HM14b] version 1; by the coarse
additive inverse property, this changes the definition of wφ by a constant depending
only on the choice of U − , U + .

5.2

Weak tiling functions on intermediate conjugacy classes

Following up on Notation A, for much of this section we shall also focus on these
additional objects whose notations we also fix:
Notation B:
• f : G → G is a CT representative of φ with core filtration element Gr representing
the free factor system F .
• The attracting lamination Λ+ corresponds to the top stratum Hs of G, an EG
stratum.
• Hsz is the union of Hs with the zero strata, if any, that it envelops, those strata
being the contractible components of Gs−1 [FH11, Definition 2.18].
• Gu = G − Hsz is the maximal filtration element that deformation retracts to Gr .
• ρs is an indivisible Nielsen path of height s if one exists (and see items (2) and (5)
under the heading “EG properties of CTs” in Section 3.1).
With this notation we have the following property:
• If ρs exists and is closed then the conjugacy class c represented by ρs fills relative
to F and so is not intermediate [HM17c, Lemma 2.5].
Our immediate goal, formulated in Proposition 5.7, is to give quantitative bounds
on well functions, expressed in terms of the CT chosen in Notation B, and more
specifically in terms of tiles as defined and studied in [BFH00, Section 3].
Earlier we recalled that a k-tile of the stratum Hs is a path of the form f#k (E)
where E is an edge of Hs and k ≥ 0. Tiles satisfy a self-similarity property saying
that for any integers l > k > 0, if the difference l − k is sufficiently large then every
l-tile contains every k-tile as a subpath [BFH00, Lemma 3.1.8 (3)].
Given a path or circuit σ in G, a k-tiling of σ is a splitting of σ each term of which
is either a k-tile or a subpath of Gs−1 with endpoints on Hs . Every generic leaf γ + of
Λ+ has a k-tiling for all k ≥ 1 [BFH00, Lemma 3.1.10 (3)].
Given a line γ in G, an exhaustion by tiles is an increasing family of finite subpaths
γ1 ⊂ γ2 ⊂ · · · whose union is all of γ such that each γi is a ki -tile for some sequence
53

k1 < k2 < · · · . Every generic leaf γ + of Λ+ has an exhaustion by tiles [BFH00,
Lemma 3.1.10 (4)].
Remark: Given a generic leaf γ + with an exhaustion by tiles γ1 ⊂ γ2 ⊂ · · · ,
those tiles define a neighborhood basis V1 ⊃ V2 ⊃ · · · of γ + where Vk ⊂ B is the
set of lines having γk as a subpath. Once an attracting neighborhood U + ⊂ B of ℓ
is chosen, the sequence φi (U + ) also forms neighborhood basis of ℓ. It follows that
for any sequence of conjugacy classes ci , the sequence of values of a well function
wφ (ci ) correlates with the value of the maximum integer ki ≥ 0 such that ci contains
a ki tile. Namely, wφ (ci ) → +∞ if and only if ki → +∞. This gives the first hint
to the quasi-comparability of well functions and of the “weak tiling functions” to be
introduced in Definition 5.6 below.
We will need a less restrictive kind of tiling, allowing terms that are Nielsen paths
of height s.
Definition 5.4 (Weak tilings). A weak k-tiling of a path or circuit σ ⊂ G is a
splitting of σ each term of which is either a k-tile, or an indivisible Nielsen path of
height s, or a maximal subpath of σ in Gs−1 . Weak k-tilings are most useful when
k is large in the following sense: the weak tiling threshold constant k0 ≥ 0 is defined
to be the minimum integer such that for all k ≥ k0 the endpoints of each k-tile are
periodic.
Examples of weak tilings: The following lemma gives general methods for constructing weak tilings. Lemma 3.1 gives a another method, with tighter quantitative
control, for producing circuits with weak 0-tilings; that lemma plays a key role in the
proof of Proposition 5.7.
Lemma 5.5. Following Notations A and B, we have:
(1) For each circuit or finite path σ ⊂ G with endpoints, if any, at vertices there
exists mσ ≥ 0 such that f#mσ (σ) has a weak 0-tiling.
(2) For all k ≥ 0, if σ = µ1 ·. . .·µI is a weak k-tiling then f# (σ) = f# (µ1 )·. . .·f# (µI )
is a weak (k + 1)-tiling.
(3) Suppose that f# (σ) = ν1 · . . . · νI is a weak (k + 1)-tiling for some k ≥ k0 and
that any endpoint of f# (σ) that is contained in Gs−1 is fixed by f (and hence
is contained in Gu ). Then there is weak k-tiling σ = µ1 · . . . · µI such that
f# (µi ) = νi for all i.
Proof. Item (1) follows from [BFH00, Lemma 4.2.6]. Item (2) follows from the fact
that if µi is a path in Gs−1 , an indivisible Nielsen path of height s, or a k-tile, then
f# (µi ) is a path in Gs−1, an indivisible Nielsen path of height s or a (k + 1)-tile
respectively.
For (3) we define µi given νi as follows. If νi is an indivisible Nielsen path of
height s, let µi = νi . Suppose next that νi is a maximal subpath of Gs−1 . An endpoint
54

of νi is either an endpoint of f# (σ) or is contained in Hs . In the former case it is fixed
by hypothesis and in the latter case it is a principal vertex by [FH11, Remark 4.9]
and hence fixed because f is rotationless. Since the components of Gs−1 − Gu are zero
strata, and since zero strata contain no fixed points, it follows that νi ⊂ Gu and that
there is a path µi ⊂ Gu with the same endpoints as νi and satisfying f# (µi ) = νi .
Suppose finally that νi is a (k + 1)-tile with endpoints say y1 and y2 . For i = 1, 2, let
xi be the unique vertex in the f -orbit of yi such that f (xi ) = yi . There exists a k-tile
µi such that f# (µi ) = νi . Since k ≥ k0 , the endpoints of µi are periodic and map to
y1 and y2 and so must be x1 and x2 . To complete the proof we need only verify that
µ1 · . . . · µI is a splitting and that adjacent terms are not both contained in Gs−1 . If
µi is an indivisible Nielsen path of height s or a k-tile then f maps the initial and
terminal directions of µi to the initial and terminal directions of f# (µi ), all of which
are contained in Hs . If µi ⊂ Gu then the initial and terminal directions of f# (µi ) are
contained in Gu . Since ν1 · . . . · νm is a splitting and since adjacent terms are not both
contained in Gu , the same is true for µ1 · . . . · µI .
Definition 5.6 (Weak tiling functions). Following notations A and B, we can now
define the integer valued weak tiling function τf on the set of intermediate conjugacy
classes. Consider an intermediate conjugacy class c that is represented by a circuit
σ ⊂ G. If c is such that σ has a weak k0 -tiling, then choose the maximal k ≥ k0 such
that σ has a weak k-tiling and define τf (c) = k − k0 . By Lemma 5.5 we can extend
−τ (c)
this to arbitrary c, by defining τf (c) to be the maximal integer such that f# f (σ)
has a weak k0 -tiling.

5.3

Coarse equivalence of well functions and
weak tiling functions.

The following proposition is the main result of this section. It states that the weak
tiling function determined by f : G → G is coarsely equivalent to any well function
associated to Λ+ . As corollaries we will state some strong regularity properties of
weak k-tilings (see Corollaries 5.10 and 5.11).
Proposition 5.7. Following notations A and B, let wφ = wφ,U be a well function for
Λ+ defined with respect to some attracting neighborhood U of a generic leaf of Λ+ ,
let k0 be the weak tiling threshold for f : G → G (Definition 5.4), and let τf be the
associated weak tiling function (Definition 5.6). Then there is a constant N so that
for all intermediate c we have
|wφ (c) − τf (c)| ≤ N
The proof of the proposition is given after the statement and proof of the following
lemma and a corollary thereof. This lemma is a relative version of [AK11, Proposition
3.8].
55

Lemma 5.8. Suppose that the lamination pair Λ± is geometric and consider ρs , the
indivisible Nielsen path of height s. There exists an integer M ≥ 1 so that if σ ⊂ G
is a circuit representing an intermediate conjugacy class then σ does not contain a
−M
subpath of the form ρM
.
s or ρs
Proof. Let c0 be the conjugacy class of the circuit in G determined by the closed
path ρs . By [HM17c, Proposition 2.18] the free factor support of c0 and F is the
same as the free factor support of Λ± and F and hence is {[Fn ]}. We may therefore
choose conjugacy classes c1 , . . . , ck carried by F whose free factor support equals
F , and therefore {c0 , c1 , . . . , ck } fills. By Whitehead’s theorem [Whi36] there is a
marked rose R such that the Whitehead graph for {c0 , . . . , ck } is connected and has
no cut vertices—this is the graph with a vertex ve for every directed edge e in R
and an edge ve —ve′ whenever ēe′ is a subpath of the circuit in R representing one of
c0 , . . . , ck or their inverses. Let τ ⊂ R be a closed path determining the circuit in R
that corresponds to c0 . It follows that if σ ′ ⊂ R is any circuit that contains τ 2 as a
subpath then the Whitehead graph for {[σ ′ ], c1 , . . . , ck } contains the Whitehead graph
for {c0 , . . . , ck } and so is also connected and has no cut points. A second application
of Whitehead’s theorem shows that {[σ ′ ], c1 , . . . , ck } is filling. To complete the proof,
apply the bounded cancellation lemma and choose M so large that if σ ⊂ G contains
a subpath of the form ρM
s then the realization of [σ] in R contains a subpath of the
2
form τ .
Corollary 5.9. For any circuit σ ⊂ G representing an intermediate conjugacy class c
the following hold:
(1) If σ has a weak k-tiling then some term of that tiling is a k-tile.
(2) For all k ≥ 0 there exists L ≥ 1 so that if σ has a weak k-tiling and if σ ′ is a
subpath of σ that crosses L edges of Hs then σ ′ contains a k-tile.
Proof. We prove (1) by contradiction, assuming that no term (of the weak k-tiling
of σ) is a k-tile. If ρs does not exist, or if it exists and no term is a copy of ρs or ρ̄s ,
then σ is contained in Gr and so c is supported by F = [Gr ], contradicting that c
is intermediate. We may therefore assume that ρs exists and that at least one term
is a copy of ρs or ρ̄s . At least one endpoint x of ρs is contained in the interior of
Hs . If ρs is not closed then σ has a term incident to x that is not a copy of ρs or ρ̄s ,
but that term must contain an edge of Hs and so can only be a k-tile, contradicting
the assumption. If ρs is closed then by a similar argument σ can only be an iterate
of ρs or ρ̄s , and so c fills relative to F (see Notation B), also contradicting that c is
intermediate.
For item (2) suppose that σ = σ1 . . . σm is a weak k-tiling. Let L0 be the maximal
number of Hs -edges in a k-tile. If ρS exists let Ls be the number of Hs edges in ρs ,
otherwise Ls = 0. If Λ± is nongeometric and ρs does not exist then the conclusion
is evident with L = 2L0 . If Λ± is non-geometric and ρs exists then ρs has distinct
56

endpoints, so if 2 ≤ i ≤ m − 1 and if σi = ρ±
s then either σi−1 or σi+1 is k-tile; the
conclusion holds with L = Ls + 2L0 . If Λ± is geometric then ρs exists and is closed
and, letting M be the constant of Lemma 5.8, at most M consecutive terms of any
cyclic permutation of the given weak k-tiling of σ can be copies of ρs or ρ̄s . In all
cases the conclusion follows with L = max{2, M} · Ls + 2L0 .
Proof of Proposition 5.7. We simplify notation slightly by writing w and τ for the
functions wφ and τf , suppressing the dependence on φ and f . Let Λ− ∈ L(φ−1 )
denote the dual repelling lamination of Λ+ .
Since w(φm(c)) = w(c) + m and τ (φm (c)) = τ (c) + m for all m ∈ Z it suffices
to show that there are uniform upper and lower bounds to τ (c) as c varies over all
intermediate conjugacy classes with w(c) = 0.
An upper bound is easy to find, using properties of Λ+ . Choose a generic leaf γ +
of Λ+ realized in G. Since U is a weak neighborhood of γ + there is subpath δ ⊂ G
of γ + such that every line in G that contains δ as a subpath is contained in U. By
[BFH00, Lemma 3.1.1 (4)] there exists kδ > 0 and a kδ -tile κ that contains δ as a
subpath. By [BFH00, Lemma 3.1.8 (4)] there exists l > max{k0 , kδ } such that every
l-tile contains every kδ -tile as a subpath. It follows that every l-tile contains δ as a
subpath. Corollary 5.9 (1) implies that any circuit that represents an intermediate c
and that has a weak l-tiling must contain an l-tile, and therefore that circuit determine
a bi-infinite line contained in U. To put it another way, if τ (c) ≥ l − k0 then w(c) ≥ 0.
It follows that w(c) < 0 =⇒ τ (c) < l − k0 and that w(c) ≤ 0 =⇒ τ (c) ≤ l − k0 , and
so l − k0 is the desired upper bound to τ (c) when w(c) = 0.
It remains to find the lower bound, using properties of Λ− . We show that there
exists d > 0 so that if w(c) = 0 and if σ ⊂ G is the circuit representing c then f#d (σ)
has a weak 0-tiling, for once that is done it follows that −(d + k0 ) is a lower bound
for τ (c). Let ζ be the line in G that wraps bi-infinitely around σ.
Claim (∗): There exists L > 0, independent of c, so that no common subpath of ζ
and a generic leaf of Λ− in G crosses L edges of Hs .
The conclusion of Claim (∗) implies that no common subpath of σ and a generic
leaf of Λ− crosses L edges of Hs , which is equivalent to the inequality ℓ−
s (σ) < L in
the hypothesis of Lemma 3.1, and so we may apply the conclusion of that lemma to
obtain d, depending only on L and so independent of c, such that f#d (σ) has a weak
0-tiling.
To prove Claim (∗), let g : G′ → G′ be a CT representing φ−1 with top stratum
′
Hs′ corresponding to Λ− and with core filtration element G′r′ ⊂ G′s′ −1 representing F .
Let γ − be a generic leaf for Λ− realized in G′ . Let σ ′ ⊂ G′ be the circuit in G′
representing c, and let ζ ′ be the line in G′ wrapping bi-infinitely around σ ′ . We shall
show:
Claim (∗′ ): There exists L′ > 0, independent of c, so that no common subpath of ζ ′
and γ − in G′ crosses L′ edges of Hs′ ′ .
57

e let H
e s be the total
This suffices to prove Claim (∗), for the following reasons. In G
es−1 = G
e\H
e s be the total lift of Gs−1 ; in G
e′ define H
e ′ ′ and
lift of Hs , and let G
s
′
e
e
Gs′ −1 similarly. Let T be the simplicial Fn -tree obtained from the universal cover G
es−1 to a point, and let T ′ be similarly obtained
by collapsing each component of G
′
′
e and G
e ′ . Using that [G′ ′ ] = [G′ ′ ] = F = [Gr ] = [Gs−1 ] it follows that
from G
r
s −1
s −1
there are equivariant quasi-isometries between T and T ′ in either direction, and that
these are equivariant coarse inverses to each other [GL07, Theorem 3.8]. For any lifts
e using subscripts to represent their realizations as lines in G
e and G
e′ and the
ζ̃, γ̃ ∈ B,
e s edges of ζ̃G ∩ γ̃G
projections to lines in T and T ′ , it follows that the number of H
equals the edge path length of ζ̃T ∩ γ̃T , which is quasi-comparable to the edge path
e ′ edges of ζ̃G′ ∩ γ̃G′ . The existence
length of ζ̃T′ ′ ∩ γ̃T−′ , which equals the number of H
s
of the uniform bound L′ in Claim (∗′ ) therefore implies the existence of the uniform
bound L in Claim (∗).
Choose an attracting neighborhood U − ⊂ B of γ − and let w ′ := wφ−1 ,U − be
the corresponding well function. By the coarse additive inverse property of well
functions we have a uniform upper bound |w ′(c) + w(c)| < W independent of c. The
set φ−W (U − ) ⊂ B is evidently also an attracting neighborhood of γ − . Using an
exhaustion by tiles of γ − , there exists an integer l ≥ 1 and an l-tile δ such that every
line having δ as a subpath is contained in φ−W (U − ). Using the self-similarity property
of tiles there exists an integer m > l such that every m-tile has δ as a subpath. Since
we are restricting our attention to those c with w(c) = 0, we obtain a uniform upper
bound w ′(c) < W . It follows that c 6∈ φ−W (U − ), and so ζ ′ does not contain δ as a
subpath. We may therefore choose L′ to be twice the maximal number of Hs′ ′ -edges
in an m-tile: since γ − has an m-tiling, every subpath of γ − that contains L′ edges
of Hs′ ′ contains an m-tile and so also contains δ, and that subpath of γ − is therefore
not a subpath of ζ ′. This completes the proof of Claim (∗′ ), of Claim (∗), and of the
lemma.

5.4

Regularity properties of weak tilings.

We continue to follow notations A and B.
A general circuit σ in G can have highly irregular behavior with respect to tiles
and weak tilings. For example, choose α to be a finite path in G having at least two
distinct illegal turns, and choose j ≥ 2 so that each k-tile with k ≥ j has more Hs
edges than α has. It follows that if k ≥ j and if σ has a weak k-tiling then σ does not
contain α as a subpath. On the other hand, one can easily construct σ so as to contain
α as a subpath and to also contain some k-tile with arbitrarily large k. This kind of
irregularity — a large gap between the greatest j for which σ has a weak j-tiling and
the greatest k for which σ contains a k-tile — is ruled out by the following corollary
for those circuits representing intermediate conjugacy classes. As an application of
the corollary, the circuit σ just constructed must fill rel F if k is sufficiently large.
58

Corollary 5.10. Following notations A and B, and letting k0 be as in Definition 5.4,
there is a positive constant M so that for all k ≥ k0 + M, every circuit σ ⊂ G
that realizes an intermediate conjugacy class and that contains a k-tile has a weak
k − M tiling.
Proof. If the corollary fails then there are sequences ki ≥ k0 and li ≥ ki such that
li − ki → +∞, and there are circuits σi ⊂ G representing intermediate conjugacy
classes, such that σi contains an li -tile but has no weak ki -tiling. Assuming this, we
argue to a contradiction.
The main step in the proof is to reduce to the case that ki = k0 for all i. More
precisely, we claim that there exist a sequence li′ → +∞ and circuits αi ⊂ G representing intermediate conjugacy classes such that αi contains an li′ -tile but has no
weak k0 -tiling. To prove the claim, decompose σi = νi νi′ as a concatenation of two
subpaths where νi is an li -tile. Let yi− and yi+ be the initial and terminal end+
points of νi respectively and let x−
i and xi be the unique periodic points satisfying
−
+
−
ki −k0
f ki −k0 (x−
(x+
i ) = yi and f
i ) = yi . There are unique paths µi connecting xi to
k
−k
k
−k
+
+
−
xi and µ′i connecting xi to xi such that f#i 0 (µi ) = νi and f#i 0 (µ′i ) = νi′ . As
argued in the proof of Lemma 5.5, µi is an (li + k0 − ki )-tile; it follows that µi has
an l tiling for all l = 1 , . . . , li + k0 − ki [BFH00, 3.1.8]. The circuit αi obtained by
tightening µi µ′i satisfies f#ki −k0 (αi ) = σi and so has no weak k0 -tiling by Lemma 5.5
and by the assumption that σi has no weak ki -tiling. Since µi is s-legal and the initial
[resp. terminal] directions of f#ki −k0 (µ̄i ) = ν̄i and f#ki −k0 (µ′i ) = νi′ are distinct, there
is a uniform upper bound to the number of Hs -edges in the maximal common initial
[resp. terminal] subpath of µ̄i and µ′i [BFH00, Lemma 4.2.2]. Thus only finitely many
edges of Hs are cancelled when µi µ′i is tightened to αi . We may therefore choose
li′′ < li so that li − li′′ is independent of i and so that at least one of the terms in the
(li′′ + k0 − ki )-tiling of the (li + k0 − ki )-tile µi remains after cancellation and so occurs
in αi . This verifies the claim with li′ = li′′ + k0 − ki .
Choose an attracting neighborhood U associated to Λ+ and let wφ be the corresponding well function. From the claim we have τf [αi ] < 0. Proposition 5.7 therefore
implies that wφ [αi ] has a uniform upper bound, and hence that there exists a positive
integer N so that for each i, αi 6∈ f#N (U). This contradicts the fact that Λ+ is a weak
limit of the αi ’s which follows from the assumption that li′ → ∞.
Corollary 5.11. Following notations A and B, let wφ be a well function for Λ+
defined with respect to some attracting neighborhood of Λ+ , and let k0 be as in Definition 5.4. Then there exists K > 0 so that for all A ≥ k0 there exist L > 0 so
that if α, β are circuits in G that represent intermediate conjugacy classes and that
have a common subpath that crosses at least L edges of Hs and if wφ ([α]) ≤ A then
wφ ([β]) ≤ A + K.
Proof. By Proposition 5.7 and Corollary 5.10 there exists an even integer K so that
so that the following are satisfied for all circuits α, β ⊂ G representing intermediate
conjugacy classes and all A ≥ k0 .
59

(1) If wφ ([α]) ≤ A then α does not contain any (A + K/2)-tiles.
(2) If wφ ([β]) > A + K then β has a weak (A + K/2)-tiling.
It therefore suffices to show that there exists L > 0 so that if β has a weak (A + K/2)tiling then every subpath of β that crosses at least L edges of Hs contains an (A+K/2)
tile. The existence of L follows from Lemma 5.9 (2).

6

Proof of Theorem E

After repeating the theorem for convenience, we shall then review some basic facts
and notations regarding the free splitting complex F S(Fn ).
Theorem E. Suppose that n ≥ 3, that H is a subgroup of IAn (Z/3) that preserves
a (possibly empty) proper free factor system F and that φ ∈ H has the following
properties:
Relative irreducibility: φ is irreducible rel F ;
Trivial restrictions: φ Out(A) is trivial for each component [A] ∈ F ;
Filling lamination: There exists a filling attracting lamination Λ+
φ ∈ L(φ), and
hence φ acts loxodromically on F S(Fn ) [HM14b].
Then φ satisfies WWPD with respect to the action of H on F S(Fn ).
−
−1
Henceforth we fix F , φ, and Λ+
φ ∈ L(φ) as in the theorem and we let Λφ ∈ L(φ )
−
be the dual repelling lamination of L(φ), and so Λφ also fills. The existence of a
filling lamination forces F to have co-edge number ≥ 2 (Fact 4.2 (3a)).

6.1

Free splitting complex and spine of relative outer space.

We first set some notation regarding the free splitting complex F S(Fn ). A free
splitting is a minimal, simplicial action of Fn y T on a simplicial tree T having trivial
edge stabilizers; usually we suppress the action from the notation and write simply
T for a free splitting. The conjugacy classes of vertex stabilizers of T forms a free
factor system F (T ). Two free splittings are equivalent if there is an Fn -equivariant
homeomorphism S 7→ T , in which case F (S) = F (T ). On each free splitting we have
the natural simplicial structure whose vertices are the points of valence ≥ 3, and we
have the natural geodesic metric which assigns length 1 to each natural edge. Fix a
set of representatives S of equivalence classes of free splittings.
The free splitting complex F S(Fn ) has a k-simplex for each T ∈ S such that the set
of natural edges has k +1 orbits under the action of Fn . The simplex corresponding to
S ∈ S has a face corresponding to T ∈ S if and only if there is a collapse map S 7→ T
60

meaning an Fn -equivariant map whose point pre-images are connected. Alternatively,
we may regard S as the vertex set of the first barycentric subdivision F S ′ (Fn ); the
1-skeleton of F S ′ (Fn ) has a directed edge from S ∈ S to T ∈ S if and only if there is a
collapse map S 7→ T ; and F S ′ (Fn ) is characterized as the directed simplicial complex
generated by its 1-skeleton, with a k-simplex for every directed path of length k.
A map f : S → T between S, T ∈ S is a function which is simplicial with respect
to some simplicial structures on S and T that refine the natural structures. Given
Φ ∈ Aut(Fn ), a map f is Φ-twisted equivariant if f (γ · x) = Φ(γ) · f (x); the case
Φ = Id yields ordinary equivariance. If not otherwise specified, maps are assumed to
be equivariant in the ordinary sense.
The canonical right action of Out(Fn ) on S is defined as follows: for each T ∈ S
and θ ∈ Out(Fn ) the image T θ ∈ S is well-defined up to equivalence by precomposing
the given action Fn y T with an automorphism Θ ∈ Aut(Fn ) representing θ. This
extends to a simplicial right action of Out(Fn ) on F S ′ (Fn ), and to a continuous action
on the Gromov bordification F S(Fn ) = F S(Fn ) ∪ ∂ F S(Fn ).
For readability, expressions of the right action T θ will sometimes be rewritten T ·θ.
Many relations between free splittings and conjugacy classes (of group elements or
subgroups) satisfy an equivariance property with respect to each θ ∈ Out(Fn ) acting
from the right on free splittings and θ−1 acting from the left on conjugacy classes. For
example, a conjugacy class c is elliptic with respect to S ∈ S if and only if θ−1 (c) is
elliptic with respect to S θ . The same format holds for relations between free splittings
and lines, and the following example of this relation will play a key role in the proof
of Theorem E:
Theorem 6.1. [HM14b, Theorem 1.2] The set of all τ ∈ ∂ F S(Fn ) which are attracting or repelling points for elements of Out(Fn ) acting loxodromically on F S(Fn )
corresponds bijectively with the set of filling laminations Λ for elements of Out(Fn ).
This correspondence is equivariant in the following manner: for all corresponding
pairs τ ↔ Λ and all θ ∈ Out(Fn ), τ θ ↔ θ−1 (Λ) is also a corresponding pair.
Let SF denote those T ∈ S such that F (T ) = F . The flag subcomplex of F S ′ (Fn )
spanned by SF is denoted KF and is called the spine of the outer space of Fn rel F ;
for example if F = ∅ then KF is the ordinary spine of the outer space of Fn . The
action of any θ ∈ Out(Fn ) takes KF to Kθ−1 (F ) , and hence we have an equation of
stabilizer subgroups Stab(F ) = Stab(KF ). The complex KF is contractible, being a
restricted deformation space in the sense of [GL07, Theorem 6.1]).
Given S ∈ SF and a conjugacy class c of Fn which is not carried by F and hence
is loxodromic on S, we say that c is S-intermediate or that c does not fill S if there
exists an Fn -orbit of edges which misses the axes of c. Note that an S-intermediate
conjugacy class is intermediate in the sense of Definition 5.1. Note also that an Sintermediate conjugacy class exists for each S ∈ SF , as follows. Since F has co-edge
number ≥ 2, there exists a finite arc α ∈ S whose endpoints v, w ∈ S have nontrivial
stabilizer, such that α misses the Fn -orbit of some edge E ⊂ S. Since edges of S
61

have trivial stabilizers, one easily produces γ ∈ Fn acting loxodromically on S with
axis equal to a union of translates of α which therefore misses the orbit of E; the
conjugacy class of γ is therefore S-intermediate.
The well function on the spine of relative outer space. Henceforth, as we
+
did in Section 5, we shall fix attracting neighborhoods of Λ−
φ and of Λφ with respect
to which the well functions wφ−1 and wφ are defined.
Consider S ∈ SF and a pair of S-intermediate conjugacy classes c1 , c2 , so for
i = 1, 2 the axis of ci in S misses the orbit of the interior of some natural edge
Ei ⊂ S. If E1 , E2 have the same orbits then c1 , c2 are supported by the same proper
free factor system — namely, the conjugacy classes of stabilizers of components of
S \ (Fn · E1 ) = S \ (Fn · E2 ) — and so we may apply [HM14b, Lemma 4.14 (3)]
to conclude that |wφ−1 (c1 ) − wφ−1 (c2 )| is uniformly bounded independent of c1 , c2 . If
on the other hand E1 , E2 have different orbits then there are one-edge free splittings
S1 , S2 connected by an edge such that ci is elliptic in Si — namely, Si is obtained
from S by collapsing to a point each component of S \ Fn · Ei — and so we may apply
[HM14b, Lemma 4.20] to get the same conclusion. Define a well function
Wφ (S) = wφ−1 (c)
for all S ∈ SF , by simply choosing c to be any S-intermediate conjugacy class. Note
that Wφ (S) is coarsely well-defined, meaning that changing the choice of c changes
the value of Wφ (S) ∈ Z by at most a constant independent of S and c. Note also that
Wφ (S) is a Lipschitz function on the spine of relative outer space: if S, T ∈ SF and
if there is a collapse map S 7→ T then any T -intermediate conjugacy class c is also
S-intermediate and so |Wφ (S) − Wφ (T )| is uniformly bounded. We record this as:
Lemma 6.2. Wφ (S) = wφ−1 (c) : SF → Z is coarsely well-defined and Lipschitz.
Remark 6.3. Here we have defined Wφ (S) for free splittings S ∈ SF where F = F (S)
is a multi-edge free splitting, whereas in [HM14b, Definition 4.15] we instead defined Wφ (T ) for one-edge free splittings T . While the domains of these two well
functions are at opposite extremes in some sense, they are nonetheless related: if
there is a collapse map S 7→ T then |Wφ (T ) − Wφ (S)| is uniformly bounded. This
is an easy consequence of [HM14b, Definition 4.15] and of the properties laid out
in Section 5.1 regarding well functions on intermediate conjugacy classes, namely
coarse well-definedness and the coarse additive inverse property; see also Remark 5.3.
One could therefore unify the two definitions to obtain a coarsely well-defined, Lipschitz well function Wφ on the free splitting complex of Fn relative to F in the sense
of [HM14a].
If c is S-intermediate and ξ ∈ H then ξ −1(c) is S ξ -intermediate. Thus
Wφ (S ξ ) = wφ−1 (ξ −1 (c))
62

for any S-intermediate c. In the special case that ξ = φm we have
m

Wφ (S φ ) = wφ−1 (φ−m (c)) = wφ−1 (c) + m
= Wφ (S) + m
(This accounts for our defining Wφ in terms of wφ−1 and not wφ ; if we had done the
m
latter, we would have Wφ (S φ ) = Wφ (S) − m.)

6.2

Setting up the proof of Theorem E.

We may assume without loss of generality that φ and φ−1 are rotationless. The action
of φ on F S(Fn ) is loxodromic by [HM14b, Theorem 1.1].
For proving Theorem E by contradiction, we assume that the action of φ is not
WWPD. From the equivalence of items (1) and (4) of Proposition 2.6, it follows that
there exists a sequence θi ∈ H satisfying the following:
Distinct Coset Property: θi−1 θj 6∈ Stab(∂± φ) for i 6= j.
Long Cylinder Property: For each S0 ∈ SF there exists R > 0 such that for any
K ≥ 0 there exists I ≥ 0 such that if i ≥ I then
d(S0 · φk θi , S0 · φk ) < R for all

0 ≤ k ≤ K.

Setting k = 0 and applying Lemma 6.2, after further increasing R the following
inequality also holds for all i
|Wφ (S0 · θi )| < R
Therefore, for any K ≥ 0 there exists I ≥ 0 such that if i ≥ I then
Wφ (S0 · φk θi ) − k < R for all 0 ≤ k ≤ K
The latter inequality uses that Wφ (S0 · φk ) = Wφ (S0 ) + k.
Henceforth we fix an initial choice of the sequence Θ = (θ1 , θ2 , . . .) satisfying the
Distinct Coset Property and the Long Cylinder Property, although notice that these
two properties each continue to hold after passing to an arbitrary subsequence of Θ,
which we will do as the proof proceeds. We also fix a choice of S0 and a corresponding
choice of R, although in later parts of the proof we may impose further constraints
on the choice of S0 (the Long Cylinder Property lets us change S0 at the expense of
increasing R).
Case analysis and outline of the proof: Let CΘ be the set whose elements are
infinite sequences (c1 , c2 , . . .) of intermediate conjugacy classes such that for some
S ∈ SF — and hence for every S ∈ SF (Lemma 6.5 (2)) — the length LS (ci ) of ci
63

in S is bounded independently of i, whereas the length LS (θi−1 (ci )) is not bounded
independently of i.
The proof of Theorem E will break into Case 1 where CΘ 6= ∅, and Case 2 where
CΘ = ∅; the geometric meaning of this case analysis is explained in Lemma 6.4. Case 2
will break into two further subcases, based on an analysis of the images of a generic
leaf of Λ− under the outer automorphisms θi−1 .
In Cases 1 and 2(a), using neither the Distinct Coset Property, nor the hypothesis
on triviality of the restrictions of φ, after passing to a subsequence of (θ1 , θ2 , . . .) we
will obtain contradictions to regularity properties of well functions that were derived
in Section 5. Having eliminated cases 1 and 2(a) we will be able to pass to a further
subsequence of (θ1 , θ2 , . . .) satisfying very strong properties. In the final case 2(b)
those properties will be combined with the hypothesis on triviality of restrictions
of φ, allowing us to pass to one final subsequence for which the images θi−1 (Λ− ) of
the repelling lamination Λ− are all the same. By applying results of [HM14b] it will
follows that the images θi (∂− φ) ∈ ∂ F S(Fn ) of the repelling fixed point are all the
same, as are the images θi (∂+ φ) ∈ ∂ F S(Fn ) of the attracting fixed point, from which
we will obtain a contradiction to the Distinct Coset Property.
For each S ∈ SF , define CΘ (S) to be set of all sequences (c1 , c2 , . . .) in CΘ such
that each ci is S-intermediate.
Lemma 6.4. The following are equivalent.
(1) CΘ = ∅.
(2) CΘ (S) = ∅ for some S ∈ SF .
(3) For all S ∈ SF , the outer automorphisms θi−1 are represented by θi−1 -twisted
equivariant maps h̄i : S → S with uniformly bounded Lipschitz constants, uniformly bounded cancellation constants, and uniformly bounded quasi-isometry
constants.
Underlying the proof of Lemma 6.4 are Lemmas 6.5 and 6.6. The first of these
includes results of Forester and of Guirardel-Levitt, together with a uniformity clause.
Lemma 6.5. For all S, T ∈ SF with their natural geodesic metrics, we have:
(1) [For02, Theorem 1.1] There exists an Fn -equivariant quasi-isometry S 7→ T .
(2) [GL07, Theorem 3.8 (7)] There exists k ≥ 1 depending only on S, T such that
for all conjugacy classes c in Fn we have k1 · LT (c) ≤ LS (c) ≤ k · LT (c).
(3) For every ℓ ≥ 1 there exists ℓ′ ≥ 1, c′ ≥ 0 such that for every Φ ∈ Aut(Fn ), each
ℓ-Lipschitz Φ-twisted equivariant map f : S 7→ T is an (ℓ′ , c′ )-quasi-isometry.

64

Proof. The twisted version of (3) follows from the untwisted version by precomposing
the action homomorphism Fn → Aut(S) with an appropriate automorphism. To
prove the untwisted version, the given map f factors as a product of an initial collapse
map—which collapses to a point each edge whose f -image is a vertex of T —followed
by a Stallings fold sequence. Since S and T have their natural geodesic metrics,
the number of folds in the sequence is bounded by a constant depending only on
the Lipschitz constant k and the rank n. It therefore suffices to consider separately
the cases that f is either a collapse map or a single fold, because a composition of
sequence of quasi-isometries has QI constants depending only on the sequence of QI
constants of the factors of the composition.
If f : S → T is a collapse map, let σ ⊂ S be the union of collapsed natural edges.
Each component of σ contains at most one natural vertex in each orbit, because
S, T have the same vertex stabilizers. The diameters of the components of σ, in the
natural metric on S, are therefore uniformly bounded by a constant D depending
only on the rank n. Given x, y ∈ S, the segment xy decomposes into an alternating
concatenation of segments of uncollapsed edges and collapsed segments of length at
1
most D, and so dT (f (x), f (y)) ≥ 1+D
d(x, y) − 2D; since f is 1-Lipschitz, this gives
uniform quasi-isometry constants.
If f : S → T is a fold map, then there are oriented natural edges E0 , E1 with the
same initial vertex v and initial segments ei ⊂ Ei with terminal points wi such that
f is a quotient map that identifies γ · e0 and γ · e1 for all γ ∈ γ. The points w0 , w1
are in different orbits and at most one has nontrivial stabilizer, because S, T have the
same vertex stabilizers. It follows that if x, y ∈ S satisfy f (x) = f (y) ∈ T then the
segment xy has length at most 4. We may assume that if ei 6= Ei then ei has length 21
and so on each edge of S the map f stretches length by a factor between 1 and 2. It
follows that f has uniform quasi-isometry constants independent of rank.
Suppose that G is a marked graph and that H is a subgraph whose non-contractible
components represent F . In this context we refer to the edges of G \ H as top edges;
since F has co-edge number ≥ 2, there are at least two top edges. We obtain a free
e by collapsing to a point each component
splitting S ∈ SF from the universal cover G
e
of the full pre-image H of H. We say that S is determined by or represented by the
graph pair (G, H). (In [HM14b] we required that H be a natural subgraph of G and
have only non-contractible components. In the current context it is more useful to
drop these requirements.) Every S ∈ SF is represented by at least one graph pair
(G, H). If S is determined by (G, H) then a conjugacy class is S-intermediate if and
only its representative in G crosses some but not all top edges.
Let SF∗ ⊂ SF be the set of those S ∈ SF such that one of two possibilities holds:
either every vertex of S has non-trivial stabilizer; or all vertex stabilizers of S are
trivial and there is only one orbit of vertices. Equivalently S is realized by a graph
pair (G, H) such that: either H is nonempty contains all the vertices of G and has no
contractible components; or H is empty and G is a rose. The first possibility applies
when F is nonempty, the second when F is empty.
65

Lemma 6.6. For each S ∈ SF∗ , for each graph pair (G, H) representing S as above,
and for each θ ∈ H, there exists a homotopy equivalence h : (G, H) → (G, H) representing θ−1 so that for each top edge E there is a closed path τE forming a circuit
and satisfying the following properties.
(1) τE crosses E but not all top edges.
(2) No top edges are cancelled when hi (τE ) is tightened to (hi )# (τE ).
(3) Each τE crosses at most two top edges, counted with multiplicity.
Proof. As a first case, assume that H contains every vertex in G and that each
component of H is non-contractible. Choose a homotopy equivalence h : (G, H) →
(G, H) representing θ that restricts to an immersion on each edge and preserves each
component of H. Let C1 and C2 be the components of H that contain the initial
and terminal endpoints of E respectively and let µ ⊂ C1 and ν ⊂ C2 be the maximal
initial and terminal subpaths of (h)# (E) that are contained in H. If C1 6= C2 then
we choose τE = Eβ2 Eβ1 where β1 ⊂ C1 and β2 ⊂ C2 are any closed paths. Top edges
are not cancelled when h(Eβ2 Eβ1 ) is tightened to (h)# (Eβ2 Eβ1 ) because ν(h)# (β2 )ν̄
and µ̄(h)# (β1 )µ do not tighten to trivial paths. If C1 = C2 then we choose τE = Eβ1
where β1 ⊂ C1 is a closed path such that ν(h)# (β1 )µ̄ does not tighten to the trivial
path. This completes the proof of (2); (1) and (3) are clear from the construction
because G \ H has at least two edges.
Suppose now that H = ∅ and that G is a rose. Choose a homotopy equivalence
h : G → G representing θ that restricts to an immersion on each edge, that fixes the
unique vertex and that has at least two gates at that vertex. The easy case is that
for each edge E, either the ends of E are contained in distinct gates or the ends of E
are contained in a single gate and there is an edge E ′ with neither end in that gate.
In the former case we take τE1 = E1 and in the latter τE1 = E1 E ′ .
The hard case is that there is a gate α such that every edge has at least one end
in α and some edge has both ends in α. We induct on the sum of the lengths |h(E)|
of the paths h(E) over all edges E that have both ends in α. Enumerate and orient
the edges of G as
E1 , . . . , EK , EK+1, . . . , EL ,

1≤K<L

so that E1 , . . . , EK have both ends in α and EK+1 , . . . , EL have only their initial end
in α.
Let β be the complement of α, consisting of the terminal ends of EK+1 , . . . , EL .
There is an oriented edge η so that the edge path h(Ej ) begins with η [resp. ends
with η̄] if and only Ej has initial end in α [resp. terminal end in α]. Post-compose
h with a map that drags the unique vertex of G across η, producing a new map
h′ : G− > G still representing θ−1 . Then the edge path h′ (Ej ) is obtained from h(Ej )
by first removing the initial η and then either removing the terminal η̄ if j ≤ K or
adding η to the terminal end if j > K.
66

Replace h : G → G by h′ : G′ → G′ . If we are now in one of the easy cases then
we are done. Otherwise there is a gate α′ such that every edge has at least one end
in α′ and some edge has both ends in α′ . All ends in the set β (the terminal ends of
EK+1 , ..., EL ) are mapped by h′ to the direction η̄, and no other end is mapped by h′
to η̄, so β is a single gate of h′ and no edge has both ends in the gate β. In particular,
β 6= α′ . Furthermore, α′ must contain all the initial ends of EK+1 . . . EL . So none of
EK+1 . . . , EL have both ends in α′ . Thus, the set of edges with both ends in α′ is a
subset of E1 , . . . , EK . Since |h′ (Ek )| = |h(Ek )| − 2 for all 1 ≤ k ≤ K, we are done by
induction.
Proof of Lemma 6.4. (3) =⇒ (1) =⇒ (2) is obvious so it suffices to show that (2) =⇒
(3). Suppose then that CΘ (S) = ∅. Once we produce the maps h̄i with uniformly
bounded Lipschtz constants, the uniform bounds on cancellation constants and quasiisometry constants follow from [BFH97, Lemma 3.1] and Lemma 6.5 respectively.
Choose S ′ ∈ SF∗ that is obtained from S by collapsing a forest. If a conjugacy class
is S ′ -intermediate then it is S-intermediate. Thus CΘ (S ′ ) = ∅. Since the equivariant
collapse map S 7→ S ′ is 1-Lipschitz, it is a quasi-isometry with uniform constants
by Lemma 6.5 and hence has an equivariant coarse inverse S ′ 7→ S with uniform
constants. It therefore suffices to verify (3) for S ′ .
Let (G, H) be a graph pair representing S ′ as in the definition of SF∗ . For each θi
in the sequence Θ and for each top edge E of G \ H, let hi : (G, H) → (G, H) be the
homotopy equivalence representing θi−1 and τE,i ⊂ G the circuit obtained by applying
Lemma 6.6. Lifting hi to the universal cover and collapsing the total lift of H, the
map hi induces a θi−1 -twisted equivariant map h̄i : S ′ → S ′ . The conjugacy class ci
determined by τE,i is S ′ -intermediate and satisfies LS ′ (ci ) ≤ 2. Since (c1 , c2 . . .) is
not an element of CΘ (S ′ ) it must be that LS ′ (θi−1 ci ) = LS ′ ([(hi )# (τE,i )]) is uniformly
bounded. Lemma 6.6 (2) implies that there is a uniform bound to the number of top
edges crossed by hi (E) for each top edge E. Equivalently, the Lipschitz constants are
uniformly bounded.

6.3

The case analysis of Theorem E.

We begin with:
Case 1: CΘ 6= ∅
It is convenient in this case to choose S0 ∈ SF∗ . Let (G0 , H0 ) be a graph pair representing S0 . For each θi in the sequence Θ and for each top edge E of G0 \ H0 , let
hi : (G0 , H0 ) → (G0 , H0 ) be the homotopy equivalence representing θi−1 and τE,i ⊂ G0
the circuit obtained by applying Lemma 6.6.
Lemma 6.7. Suppose that (c1 , c2 , . . .) is an element of CΘ . Then there are arbitrarily
large i and arbitrarily large M and a top edge E of G0 such that hi (E) and the circuit
representing θi−1 (ci ) have a common subpath that crosses M top edges.
67

Proof. Decompose the circuit σi ⊂ G0 representing ci into a concatenation of a uniformly bounded number of subpaths, each of which is either contained in H or is
a top edge. The circuit (hi )# (σi ) representing θi−1 (ci ) therefore decomposes into a
concatenation of a uniformly bounded number of subpaths, each of which is either
contained in H or contained in the hi -image of some top edge. Since (c1 , c2 . . .) ∈ CΘ ,
the number of top edges crossed by the latter subpaths is unbounded.
m

For each m ≥ 0, let Sm = S0φ . By Lemma 6.4 we can choose an element
(c1 , c2 , . . .) in C(Sm ). From the fact that ci is Sm -intermediate, it follows that θi−1 (ci )
θi
is Sm
-intermediate. Thus
θi
Wφ (Sm
) = wφ−1 (θi−1 (ci ))
for all i. Similarly, Lemma 6.6(1) implies that
Wφ (S0θi ) = wφ−1 ([(hi )# (τE,i )])
where [(hi )# (τE,i )] is the conjugacy class of the circuit formed by (hi )# (τE,i ). By
Lemma 6.7, we can choose i and a top edge E of G0 so that the circuit representing θi−1 (ci ) in G0 contains a subpath of hi (E) that crosses at least m top edges.
Lemma 6.6(2) then implies that the circuit representing θi−1 (ci ) in G0 and the circuit
(hi )# (τE,i ) have a common subpath that crosses at least m top edges. Since there is
a uniform bound for wφ−1 ([h# (τE,i )]) = Wφ (S0θi ), Corollary 5.11 implies that there is
θi
). This contradicts the assumption that
a uniform bound for wφ−1 (θi−1 (ci )) = Wφ (Sm
θi
|Wφ (Sm
) − m| < R for all sufficiently large i and so completes the proof in case 1.
Case 2: C = ∅
Let f : G → G be a CT representing φ−1 with top stratum Gs and with F realized
by a core filtration element Gr . Let Hsz be the union of Hs with the zero strata, if
any, that it envelops [FH11, Definition 2.18] and let Gu = G − Hsz . Each zero stratum
enveloped by Hs is a contractible component of Gs−1 . Since φ [Gr ] is trivial and φ is
irreducible rel [Gr ] ⊏ [Gs ], f Gr is the identity and Gu deformation retracts to Gr .
The edges of Gu − Gr are non-fixed NEG and so are linear.
We now constrain S0 ∈ S to be the free splitting determined by the graph pair
(G, Gs−1). By Lemma 6.4, there exist θi−1 twisted equivariant maps h̄i : S → S with
uniformly bounded Lipschitz constants, cancellation constants, and quasi-isometry
constants. Letting hi : (G, Gs−1 ) → (G, Gs−1 ) be homotopy equivalences corresponding to h̄i : S → S, it follows that:
(1) For each top edge Ej the number of top edges in hi (Ej ) is uniformly bounded.
(2) There is a uniform constant C so that if σ = σ1 σ2 ⊂ G is a decomposition
into subpaths, then for all i at most C pairs of top edges are cancelled when
(hi )# (σ1 )(hi )# (σ2 ) is tightened to (hi )# (σ).

68

(3) For all L there exists L′ independent of i so that if σ ⊂ G crosses ≥ L′ top
edges then (hi )# (σ) crosses ≥ L top edges.
(4) For all L′ there exists L independent of i so that for any path σ ⊂ G, if (hi )# (σ)
crosses ≥ L top edges then σ crosses ≥ L′ top edges.
As a consequence of (2) we have the following consequence for the path maps
(hi )## (see Section 3.1 or [HM17c, Section 1.1.6] for a more comprehensive review):
(5) For all i and all paths σ ⊂ G, (hi )## (σ) contains the path obtained from
(hi )# (σ) by removing the maximal initial and terminal subpaths with exactly
C top edges [HM17c, Lemma 1.6].
After passing to a subsequence of the θi ’s (and the corresponding subsequences of
ni ’s and hi ’s), we may assume by (1):
(6) For each top edge Ej the sequence of top edges crossed by hi (Ej ) is independent
of i.
Given a top edge Ej , write hi (Ej ) = α0 ǫ1 α1 . . . ǫk αk where ǫ1 , . . . , ǫk are the top
edges that are crossed. After passing to a further subsequence of the θi ’s we may
assume that for all 0 ≤ l ≤ k, the subpath αi is either always trivial or never trivial,
independent of i. We may therefore subdivide Ej into subpaths called edgelets and
modify hi , without changing hi (Ej ), so that:
(7) For all i and j, hi maps each edgelet in Ej to either a top edge or to a non-trivial
path in Gs−1 .
Recall that f : G → G represents φ−1 , not φ, and that tiles are defined with respect to
f . As k goes to infinity, the number of top edges in a k-tile goes to infinity. We may
therefore choose a fixed integer K ≥ 1 so that for each i, j, the K-tile ωj = f#K (Ej )
has the property that its (hi )# -image σij := (hi )# (ωj ) crosses at least 3C top edges.
There may be some cancellation of top edges when hi (ωj ) is tightened to σj . After
passing to a further subsequence of the θi ’s, we may assume that this cancellation is
independent of i; in particular, for each K-tile ωj , the sequence of top edges crossed
by σij is independent of i. Decompose σij into subpaths σij = σij− σij+ where σij− is the
shortest initial subpath of σij that completely contains C top edges and that terminates at the midpoint of a top edge. Note that the endpoints of σij− are independent
of i. For each K-tile ωj we define its prefix–suffix decomposition ωj = ωj− ωj+ where
ωj− is the shortest initial subpath satisfying (hi )# (ωj− ) = σij− . The common endpoint
of ωj− and ωj+ , which we refer to as the midpoint of ωj , is contained in the interior of
an edgelet and maps to the common endpoint of σij− and σij+ . Since the cancellation
of top edges when hi (ωj ) is tightened to σj is independent of i we have:
(8) For each K-tile ωj its prefix–suffix decomposition ωj = ωj− ωj+ , its midpoint and
the hi -image of its midpoint are independent of i.
69

A generic leaf γ − of Λ−
φ has a 0-tiling; i.e. a splitting whose terms are either
edges in Hs or paths in Gs−1 . After increasing K if necessary, we may assume that
f K (Hj ) ⊂ Gu for each zero stratum Hj enveloped by Hs . Applying f#K to the above
0-splitting, we have a splitting of γ − whose terms are either K-tiles or paths in Gu .
We may therefore write γ − as an alternating concatenation of the form
γ − = · · · ωjt−1 ρt ωjt ρt+1 ωjt+1 · · ·
where each ωj is a K-tile and each ρt is either trivial or contained in Gu . After taking
the prefix–suffix decomposition of each K-tile and then collecting terms we obtain a
new decomposition of γ − as follows:
(∗)

γ − = · · · ωj−t−1 ωj+t−1 ρt ωj−t ωj+t ρt+1 ωj−t+1 ωj+t+1 · · ·
| {z } |
{z
}
µt

µt+1

= · · · µt−1 µt µt+1 µt+2 · · ·

Our choice of C guarantees that the above decomposition is a “universal 1-splitting”
in the sense that
(hi )(γ − ) = . . . (hi )# (µ−1 ) (hi )# (µ0 ) (hi )# (µ1 ) . . .
is a decomposition into subpaths for all hi . By (8),
(9) For each µt , the hi -image of the endpoints of µt are independent of i.
The proof now divides into two subcases.
Case 2a: There exists p < q so that for infinitely many values of i, the
path (hi )# (µp . . . µq ) is not a subpath of γ − .
After passing to a subsequence of the θi ’s we may assume that the assumption of
Case 2a holds for all values of i. Choose p′ < p and q ′ > q so that (hi )# (µp′ . . . µp−1)
and (hi )# (µq+1 . . . µq′ ) cross at least C top edges. By (5)
(hi )## (µp′ . . . µq′ ) ⊃ (hi )# (µp . . . µq )
for all i. By [BFH00, Lemma 3.1.8] there exists K ′ such that
′

f#K (Ej ) ⊃ µp′ . . . µq′
for each j. From [HM17c, Lemma 1.6-(3)] it therefore follows that
′

(hi )## (f#K (Ej )) ⊃ (hi )## (µp′ . . . µq′ )
We conclude that:
70

(#) If a path σ0 contains a K ′ -tile then (hi )# (σ0 ) contains (hi )# (µp . . . µq ), and
hence (hi )# (σ0 ) does not occur as a subpath of γ − .
Choose a circuit α ⊂ G that crosses some but not all edges of G − Gs−1 . Then
the circuit
βi,m := (hi )# f#m (α) ⊂ G
determines a conjugacy class in Fn that is represented by a circuit in G · φm θi which
crosses some but not all top edges of (G, Gs−1) · φm θi . It follows that
wφ−1 [βi,m ] = W (S0 · φm θi )
and hence (after further increasing R) for each m we have |wφ−1 [βi,m ] − m| < R if i
is sufficiently large. It follows that:
(##) There is a sequence Lm → +∞ such that for each m, if i is sufficiently large
then there exists a common subpath of γ − and βi,m that crosses Lm top edges.
For reasons as above we have
m

wφ−1 [f#m (α)] = W (S0φ )
Thus wφ−1 [f#m (α)] goes to infinity with m. By Proposition 5.7, for all sufficiently large
m, f#m (α) has a weak K ′ -tiling. By Lemma 5.9 (2) there exists L′ > 0 so that for
all sufficiently large m, every subpath σ ⊂ f#m (α) that contains at least L′ top edges
contains a K ′ -tile. Choose L > 0 so that if σ is any path in G and if ζ is a subpath of
(hi )# (σ) that crosses at least L top edges then there is a subpath σ0 of σ that crosses
at least L′ top edges such that (hi )# (σ0 ) is a subpath of ζ; this follows by applying
item (4) above, with our current ζ in the role of the σ of item (4). Thus any subpath
of βi,m that crosses at least L top edges contains a subpath of the form (hi )# (σ0 )
where σ0 is a subpath of f#m (α) that contains at least L′ top edges, and hence σ0
contains a K ′ -tile. Combining this with conclusion (#), it follows that no subpath of
βi,m that crosses at least L edges occurs as a subpath of γ − . This contradicts (##),
completing the proof in Case 2a.
Case 2b: For any p < q, the path (hi )# (µp . . . µq ) is a subpath of γ − for all
sufficiently large i.
Our first claim is that
(∗) For any given p, as i varies, the expression (hi )# (µp ) takes on only finitely many
values.
We can see this as follows. The number of top edges in (hi )# (µp ) is uniformly bounded
so the sequence of top edges in (hi )# (µp ) takes on only finitely many values. We
are therefore reduced to bounding the number of edges in a maximal subpath ν of
(hi )# (µp ) in Gs−1 . Since γ − is birecurrent, there exists q > p such that µp = µq .
71

There is a copy ν ′ of ν in (hi )# (µq ) that is separated from ν in (hi )# (µp . . . µq ) by a
uniformly bounded number of top edges. Assuming without loss that (hi )# (µp . . . µq )
is a subpath of γ − , we have that ν and ν ′ are separated in γ − by a uniformly bounded
number, say P , of top edges. Choose k so that every k-tile contains at least P + 1
edges. Since γ − decomposes into k-tiles and subpaths in Gs−1 , at least one of ν and
ν ′ must be contained in a k-tile. This gives the uniform bound on the number of
edges in ν and so completes the proof of the first claim.
The topological structure of Λ−
φ . The fact that Gu is the union of a filtration
element of fixed edges Gr and some NEG-linear edges Gu \ Gr implies that Λ−
φ has
a particularly simple topological structure. If there were no linear edges, then Λ−
φ
would be minimal, meaning every leaf would be dense. With only linear edges, the
structure is still quite simple: for example, the only nondense leaves are the finitely
many periodic lines corresponding to circuits around which the linear edges of Gu \Gr
twist. We now describe a decomposition of generic leaves of Λ−
φ which reflects this
structure.
The generic leaf γ − decomposes as a concatenation of subpaths in Hs and maximal
subpaths in Gs−1. We further divide the latter into subpaths in zero strata enveloped
by Hs called middle pieces (referred to in [FH11] as ‘taken connecting paths’) and
subpaths in Gu called bottom pieces. There are only finitely many middle pieces and
the f# -image of a middle piece is either another middle piece or a bottom piece. Each
tile also divides into subpaths in H, bottom pieces and middle pieces. A bottom piece
is primitive if it occurs in a 1-tile or is the f# -image of a middle piece. There are
only finitely many primitive bottom pieces; we denote them with the symbol δ. Every
bottom piece of a k-tile can be written as f#j (δ) for some primitive bottom piece δ
and some j ≥ 0. This is obvious for k = 1 and it follows by the obvious induction
argument on k using the fact that each k-tile is s-legal.
Recall that f Gr is the identity and that each edge of Gu \ Gr is NEG-linear.
Since the initial vertices of edges of Gu − Gr have valence one in Gu , paths in Gu can
only cross edges of Gu − Gr in their first or last edges. A primitive bottom piece δ
therefore has one of three types:
Constant Type: δ is contained in Gr ;
Once-Linear Type: Either the initial or terminal edge of δ, but not both, is contained in Gu − Gr ;
Twice-Linear Type: The initial and terminal edges of δ are contained in Gu − Gr .
If δ has constant type then f#j (δ) is independent of j. If δ has once linear type then
there is an edge E of Gu − Gr , a closed path u in Gr such that f#j (E) = Euj , and a
path β in Gr , such that
δ = E β, and f#j (δ) = E [uj β]
or δ = β E, and f#j (δ) = [β ūj ] E
72

If δ has twice linear type then there are edges E, E ′ in Gu − Gr , closed paths u, u′ in
Gr such that f#j (E) = Euj and f#j (E ′ ) = E ′ u′j , and a path β in Gr , such that
δ = E β E′

and f#j (δ) = E [uj β ū′j ] E

′

Recall that for each term µt of the decomposition γ − = · · · µt−1 µt µt+1 µt+2 · · ·
there is a decomposition µt = ωj+t−1 ρt ωj−t where ωj+t−1 is a K-tile suffix, ρt is either
trivial or a maximal subpath in Gu , and ωj−t is a K-tile prefix. Since every non-trivial
subpath of γ − occurs in a tile, each non-trivial ρt can be written as f#j (δ) for some
primitive bottom piece δ and some j ≥ 0.
To each K-tile suffix ω + , each primitive bottom piece δ, and each K-tile prefix
−
ω we associate the set of paths
M(ω +, δ, ω − ) = {µt

ω + = ωj+t−1 , and f#j (δ) = ρt for some j ≥ 0, and ω − = ωj−t }

Also, for each K-tile suffix ω + and K-tile prefix ω − we define
(
{ω + ω − } = {µt }
if ω + = ωj+t−1 and ω− = ωj−t for some t
+ −
M(ω , ω ) =
∅
otherwise
We say that the set M(ω +, δ, ω − ) has constant, once-linear or twice-linear type if
δ does. The set {µt : t ∈ Z} is the (not necessarily disjoint) union of the sets
M(ω +, δ, ω − ) and the sets M(ω +, ω − ) as ω + , ω − , and δ vary. Note that there are
finitely many of the sets M(ω +, δ, ω − ) and M(ω +, ω − ). Note also that each set
M(ω +, ω − ) has finite cardinality whereas the sets M(ω +, δ, ω − ) may have infinite
cardinality.
Our second claim is that:
(∗∗) Each M(ω +, δ, ω − ) contains a finite subset M0 (ω +, δ, ω − ) so that for any i and j,
if (hi )# (µt ) = (hj )# (µt ) for all µt ∈ M0 (ω +, δ, ω − ) then (hi )# (µt ) = (hj )# (µt )
for all µt ∈ M(ω +, δ, ω − ).
Before proving this claim we use it to complete the proof of Theorem E, by arriving
at a contradiction that settles Case 2b. From claim (∗) it follows that for any finite
subset M ⊂ {µt : t ∈ Z} we can pass to a subsequence so that (hi )# (µt ) = (hj )# (µt )
for all i, j and for all µt ∈ M. We may therefore assume that (hi )# (µt ) = (hj )# (µt ) for
all i, j and for all µt in each M(ω − , ω + ) and in each M0 (ω + , δ, ω − ). From claim (∗∗) it
then follows that (hi )# (µt ) = (hj )# (µt ) for all i, j and all µt . This in turn implies that
(hi )# (γ − ) = (hj )# (γ − ) for all i, j, and hence taking weak closures that θi−1 (Λ− ) =
θj−1 (Λ− ). It follows by [HM14b, Theorem 1.2] that θi (∂− φ) = θj (∂− φ) and that
θi (∂+ φ) = θj (∂+ φ), and so θi−1 θj (∂± φ) = ∂± φ, in contradiction to the Distinct Coset
Property saying that θi−1 θj 6∈ Stab(∂± φ).
It remains to verify claim (∗∗). This is obvious if M(ω +, δ, ω − ) is finite, in particular if it has constant type and therefore has cardinality at most one. For notational
73

simplicity we let M = M(ω +, δ, ω − ), we assume M is infinite, and we denote the
desired finite subset by M0 = M0 (ω +, δ, ω − ). There exists an infinite sequence of
integers 0 ≤ j1 < j2 < . . . so that
M = {τl := ω + f#jl (δ) ω −

l ≥ 1}

For all l ≥ 1, the paths τl have the same initial vertex v− and the same terminal
′
′
vertex v+ . The images v−
:= hi (v− ) and v+
:= hi (v+ ) are independent of i by (9).
+ j1
−
We include τ1 = ω f# (δ) ω in M0 , and then applying (∗) we pass to a subsequence
of the θi ’s so that hi (τ1 ) is independent of i.
For each l ≥ 2 consider the closed edge path τ1 τ̄l based at v− , with nontrivial path
homotopy class denoted [τ1 τ̄l ] ∈ π1 (G, v− ). Let γl denote the closed path obtained
by straightening τ1 τ̄l , and so γl is the unique closed path based at v− that represents
[τ1 τ̄l ]. Let D = {[τ1 τ̄l ] l ≥ 2} ⊂ π1 (G, v− ), an infinite set of nontrivial elements of
π1 (G, v− ).
′
Let Υi : π1 (G, v− ) → π1 (G, v−
) be the isomorphism induced by (hi )# . Under the
identification π1 (G, v− ) ≈ Fn , we obtain automorphisms
Υ−1
i Υ1 ∈ Aut(π1 (G, v− )) ≈ Aut(Fn )
Denote their fixed subgroups by
−1
Bi = Fix(Υ−1
i Υ1 ) = Fix(Υ1 Υi ) < Fn

For each l ≥ 2, it follows that the path (hi )# (τl ) is independent of i if and only if
′
Υi [τ1 τ̄l ] ∈ π1 (G, v−
) is independent of i if and only if [τ1 τ̄l ] ∈ Bi for all i.
Each Bi has rank ≤ n by the solution to the Scott conjecture [BH92, Theorem
6.1], and each Bi is primitive meaning that if Bi contains a nonzero power of an
element then it contains that element.
It therefore suffices to show that there exists a finite subset D0 ⊂ D such that for
any subgroup B < Fn that is a primitive and has rank ≤ n, if D0 ⊂ B then D ⊂ B.
Once that is shown then we finish claim (∗∗) by defining M0 = {τl [τ1 τ̄l ] ∈ D0 }.
For the rest of the proof we fix B < Fn to be primitive and of rank ≤ n; we shall
define the subset D0 ⊂ D independent of B.
If the set M is of once-linear type then either δ = Eβ or δ = β Ē, and f#j (E) =
Euj . The two cases are symmetric so we assume that δ = Eβ. Thus
τl = ω + E [ujl β] ω −
γl = (ω + E) uj1−jl (ω + E)−1
It follows that as l varies, the [τ1 τ̄l ]’s are all contained in a cyclic subgroup of π1 (G, v− ).
Any primitive subgroup that contains a single nontrivial element of this cyclic subgroup contains all of D, so taking D0 to be any nonempty, finite subset of D we are
done in the once-linear case.
74

We are now reduced to the case that M is of twice-linear type. In this case we
′
have δ = E β E , f#j (E) = Euj , and f#j (E ′ ) = E ′ u′j , and so
′

τl = ω + E [ujl β ū′jl ] E ω −
γl = (ω + E) [uj1 β (u′)jl −j1 β̄ u−jl ] (ω + E)−1
If the number of edges cancelled when uj1 β (u′)jl −j1 β̄ u−jl is tightened is not bounded
independently of l then there is a closed path α such that u′ , β and u are all iterates
of α. The path uj1 β(u′ )jl −j1 β̄u−jl therefore tightens to an iterate of α and so each
γl has the form (ω + E) α∗ (ω + E)−1 . Each [τ1 τ̄l ] is therefore contained in the cyclic
subgroup of π1 (G, v− ) generated by (ω + E) α (ω + E)−1 , and the proof concludes as in
the once linear case with D0 any nonempty, finite subset of D.
It remains to consider the case that the cancellation of uj1 β (u′)jl −j1 β̄ u−jl is uniformly bounded. In this case, there are paths σ1 , σ2 , σ3 , root-free closed paths α1 , α2 ,
an integer l0 ≥ 2, and increasing sequences of positive integers {s(l)} and {t(l)}
defined for l ≥ l0 , such that if l ≥ l0 then γl can be written in the form
s(l)

−t(l)

γl = σ1 α1 σ2 α2

σ3

and such that the following hold:
• u′ is an iterate of α1 , and u is an iterate of α2 ;
• The path σ1 is obtained by straightening the concatenation of ω + E uj1 β followed by some initial subpath of (u′ )∗ ;
• The path σ2 is obtained by straightening the concatenation of some terminal subpath of (u′ )∗ , followed by the path β̄, followed by some initial subpath
of (u−1 )∗ ;
• The path σ3 is obtained by straightening the concatenation of some terminal
subpath of (u−1 )∗ followed by E ω̄ + .
• The maximal common initial subpath of σ2 and α1 is a proper initial subpath
of α1 ; we denote this subpath σ2′ .
Let
D0 = {[τ1 τ̄l ] 2 ≤ l ≤ l0 + 2n − 1} ⊂ π1 (G, v− )
Assuming that D0 ⊂ B we complete the proof by showing that the path σ1 α1s σ2 α2t σ3
represents an element of B for all s, t ∈ Z.
Since B < π1 (G, v− ) has rank ≤ n, there is a finite core graph X of rank ≤ n
equipped with a basepoint x and an immersion Q : (X, x) → (G, v− ) such that a closed
path in G based at v− lifts to a closed path in X based at x if and only if it represents
an element of B. Equivalently, the induced homomorphism Q∗ : π1 (X, x) → π1 (G, v− )
75

is an injection with image B. The path γl0 +2n−1 representing [τ1 τ̄l0 +2n−1 ] ∈ D0 lifts to a
closed path γ̃l0 +2n−1 in X based at x. We have the following forms for a decomposition
of γl0 +2n−1 and the corresponding decomposition of γ̃l0 +2n−1 :
γl0 +2n−1 = σ1 α1 α1 · · · α1 σ2 α2−1 α2−1 · · · α2−1 σ3
{z
}
|
|
{z
}
s(l0 + 2n − 1) times

t(l0 + 2n − 1) times

−1
−1
−1
γ̃l0 +2n−1 = σ̃1 α̃1,1 α̃1,2 . . . α̃1,s(l0 +2n−1) σ̃2 α̃2,1
α̃2,2
. . . α̃2,t(l
σ̃3
0 +2n−1)

In X denote the following vertices, where 1 ≤ i < s(l0 + 2n − 1) − 1:
y0 = the terminal vertex of σ̃1 = the initial vertex of α̃1,1
yi = the terminal vertex of α̃1,i = the initial vertex of α̃1,i+1
′
σ̃2,i = the unique lift of σ2′ with initial vertex yi
′
zi = the terminal vertex of σ̃2,i
′
Also, for l0 ≤ l ≤ l0 + 2n − 2, the initial and terminal vertices of σ̃2,s(l)
are denoted
ηl = ys(l) and let ζl = zs(l) .
The vertex ζl0 has valence ≥ 3 in X, because both of the paths γl0 and γl0 +1 lift
to paths in X starting at x, these two lifts have maximal common initial subpath
′
σ̃1 α̃1,1 . . . α̃1,s(l0 ) σ̃2,s(l
, this subpath is proper in each of γl0 and γl0 +1 , and the ter0)
minal endpoint of this subpath is ζl0 ; this follows by maximality of σ2′ . Similarly each
of the vertices ζl0 +1 , . . . , ζl0 +2n−2 has valence ≥ 3 in X.
Since X has at most 2n − 2 distinct vertices of valence ≥ 3, it must be that there
is repetition amongst the vertices ζl0 , . . . , ζl0 +2n−2 . Let a < b be indices such that
ζla = ζlb . It follows that ys(la ) = ηla = ηlb = ys(lb ) . Since there is at most one lift
of α1 to X that terminates at each yi , it must be that y0 = ys(lb )−s(la ) . This implies
that some iterate of α1 lifts to a closed path in X based at y0 and hence that some
straightened iterate of σ1 α1 σ̄1 lifts to a closed path in X based at x. Since B is
primitive, the path σ1 α1 σ̄1 itself lifts to a closed path based at x and so determines
an element of B.
The symmetric argument shows that σ̄3 α2 σ3 determines an element of B and hence
that each path

σ1 α1s σ2 α2t σ3 = [(σ1 α1s−s2 σ̄1 )(σ1 α1s2 σ2 α2−t2 σ3 )(σ̄3 α2t+t2 σ3 )]
determines an element of B for each s, t ∈ Z as needed to complete the proof.

7

Appendix: Proof of Theorem D

This proof hews closely to the proof of [BF02] Theorem 8. The one innovation is to
use the Kaloujnin–Krasner embedding to avoid the wreath product hypothesis in the
statement of [BF02] Theorem 8. This results in many slight technical differences from
76

[BF02], and so we have written the proof to be primarily self-contained, but with a
few references to [BF02] and to [Fuj98]. We begin by setting up the notation of the
Kaloujnin–Krasner embedding which is used throughout the proof. Substantial use
of algebraic properties of the embedding comes towards the end of the proof.
Fix a group Γ, a finite index normal subgroup N ⊳ Γ, an action N y X on a
hyperbolic complex X, and a rank 2 free subgroup F < N, with respect to which the
global WWPD hypothesis of Definition 2.13 holds: each element of N is either loxodromic or elliptic; the restricted action F y X is a Schottky group whose nontrivial
elements are all WWPD elements of the action N y X; and for each inner automori
phism i : G → G, the restriction to F of the composed action N −
→ N y X is either
elliptic or is a Schottky group whose nontrivial elements are all WWPD elements of
the composed action. We shall prove that Hb2 (Γ; R) contains an embedded ℓ1 .
We may assume that F < [N, N], because there is a natural inclusion of commutator subgroups [F, F ] < [N, N], and because the global WWPD hypothesis is retained
when F is replaced by any of its rank 2 subgroups, so we may replace F with a rank 2
subgroup of [F, F ]. This assumption will be used only at the very last sentence of the
proof.
Step 1: The Kaloujnin–Krasner embedding. This is an embedding of Γ into a certain
wreath product constructed from N ⊳ Γ. Consider the quotient group Q = Γ/N =
{B = Ng g ∈ Γ}. Elements of the wreath product N Q ⋊ Sym(Q) are denoted
as ordered pairs (ρ, φ) ∈ N Q × Sym(Q), the group operation being (ρ, φ) · (σ, ψ) =
(ρ · (σ ◦ φ), ψφ); note the reversal of order in the second coordinate. The embedding
θ : Γ ֒→ N Q ⋊Sym(Q) is given by the following formula.1 For each coset B ∈ Γ/N = Q
choose a representative gB ∈ B, and so B = NgB ; in particular, choose gN to be the
identity. For each µ ∈ Γ define θ(µ) = (ρµ , φµ ) ∈ N Q ⋊ Sym(Q) where for each B ∈ Q
we have:
−1
ρµ (B) = gB µ gBµ
∈N

φµ (B) = Bµ ∈ Q
One may easily check this to be a homomorphism. To see that it is injective, note
first that φµ is trivial if and only if B = Bµ for all B ∈ Γ/N if and only if Nν = Nνµ
for all ν ∈ Γ if and only if µ ∈ N (we shall need this below). Next, if µ ∈ N and if
−1
ρµ (B) is trivial for all B ∈ Q, then using B = N it follows that ρµ (N) = gN µ gN
µ =
−1
gN µ gN = µ is trivial.
This completes Step 1.
We next set up some notation that will be used for the rest of the proof of Theorem 2.5. In particular we match the notation regarding the Kaloujnin–Krasner
embedding with our notation for outer representatives of the action of Γ on N.
1

See e.g. [Wel76], proofs of Proposition 12.1 and Theorem 11.1

77

Under the natural injection N Q ֒→ N Q ⋊Sym(Q) given by ρ 7→ (ρ, Id), we identify
N with its image. Under this identification, note that θ(Γ) ∩ N Q = θ(N) because,
as just seen, φµ is trivial if and only if µ ∈ N.
Fix an enumeration of Q = Γ/N as N = B1 , . . . , BK , and abbreviate gBγ to gκ . For
each κ = 1, . . . , K consider the map iκ : N → N defined by the following composition:
Q

iκ

N

/
θ

θ(N)

θ(Γ) ∩ N Q

⊂

/

NQ

)/
ρ7→ρ(Bκ )

N

Tracing through the definitions, and using that if µ ∈ N then Bµ = B, it follows that
iκ : N → N is the outer representative of the action of Γ on N determined by gκ :
iκ (µ) = ρµ (Bκ ) = gκ µ gB−1κµ = gκ µ gκ−1
In particular, since N = B1 is represented by g1 = Id the automorphism i1 is the
identity and the N y1 X is the given action N y X; we therefore often drop the
subscript 1 from the action N y1 X.
Since each action F yκ X is either Schottky or elliptic, and since F y1 X is
Schottky, after re-indexing there exists K1 ≥ 1 such that F yκ X is Schottky when
1 ≤ κ ≤ K1 and it is elliptic when K1 < κ ≤ K.
Consider the equivalence relation ∼ on elements of the action N y X, as defined
in Section 2.3. We let ∼κ denote the analogous equivalence relation on the elements
of each action N yκ X, which is given by
µ ∼κ ν ⇐⇒ iκ (µ) ∼ iκ (ν)
Step 2: A good free basis for F . We may assume, after replacing F with a rank 2 free
subgroup, that there is a free basis F = hg1 , h1 i such that for each κ = 1, . . . , K we
have
(1) g1 6∼κ h1 .
(2) g1 6∼κ h−1
1 .
To see why, start with an arbitrary free basis F = hg1 , h1 i, and suppose by induction
that we have k ∈ {1, . . . , K} such that (1) and (2) hold for 1 ≤ κ < k (which
is vacuously true when k = 1). If k > K1 then F yk X is elliptic and so (1)
and (2) hold as well for κ = k. We may therefore assume that κ ≤ K1 . The
subgroup hg1 i is a maximal cyclic subgroup of F , since g1 is a free basis element.
It follows that for each of the actions F yκ X, 1 ≤ κ ≤ k, the subgroup of F
that stabilizes the fixed point pair of the loxodromic isometry g1 yκ X is equal to
hg1 i, and therefore h1 does not stabilize the fixed point pair of g1 for any of these
actions. Applying Proposition 2.11 we obtain for each 1 ≤ κ ≤ k a constant Aκ
78

such that if a ≥ Aκ then g1 and h′1 = g1a h1 g1−a h−1
satisfy properties (1) and (2).
1
Taking a ≥ max{Aκ 1 ≤ κ ≤ K1 } and replacing h1 with h′1 , those properties hold
simultaneously for all κ ≤ K1 . Having made this replacement, the WWPD hypothesis
for Γ with respect to N y X and F remains true, completing Step 2.
We shall often use the above “maximization trick” to obtain constant bounds
independent of κ.
Some notations: Let T denote the Cayley tree of the free group F = hg1 , h1 i.
For each κ ≤ K1 we have a continuous, Kκ , Cκ quasi-isometric embedding ξκ : T → X
which takes vertices to vertices and edges to edge paths and which is F -equivariant
with respect to the action N yκ X. The quasi-isometry constants Kκ ≥ 1, Cκ ≥ 0
of the maps ξκ at first depend on κ, but applying the maximization trick we have:
(3) There exists K ≥ 1, C ≥ 0 independent of κ such that ξκ : T → X is a K, C
quasi-isometric embedding for κ ≤ K1 .
Fix δ to be a hyperbolicity constant for X. Given a nontrivial g ∈ F , let AT (g)
denote its axis in the tree T , along which the translation distance of g denoted Lg
equals the cyclically reduced word length of g. For each κ ≤ K1 , let Aκ (g) = ξκ (AT (g))
which is a K, C-quasi-axis of g with respect to the Schottky action F yκ X. By
item (3), the quasi-axes Aκ (g) all have uniform quasi-isometry constants K ≥ 1,
C ≥ 0, independent of κ ≤ K1 and g ∈ F .
Again we often drop the subscript κ = 1, and so A(g) = A1 (g) is the K, C-quasiaxis with respect to the given action N y X = N y1 X.
Step 3: A sequence of inequivalencies. There exists a sequence of nontrivial elements
f1 , f2 , f3 , . . . ∈ F < [N, N]
such that the following inequivalence properties hold:
(4) For each i ≥ 1 and each 1 ≤ κ ≤ K we have fi 6∼ iκ (fi−1 ).
(5) For each i > j ≥ 1 and each 1 ≤ κ ≤ K we have fi 6∼ iκ (fj± ).
If κ > K1 then iκ (fi−1) and iκ (fj±1 ) are both elliptic and are therefore distinct from
the loxodromic element fi , hence (4) and (5) both follow from Definition 2.9. We may
therefore assume that κ ≤ K1 . In that case the justifications we give will follow [BF02]
Proposition 2, Claim 3 together with lines of argument found in [BF02] Theorem 8,
on pages 81–82 (here we hew more closely to the notation of [BF02]).
The elements fj are given by formulas
−s

−t

m

n

k

−lj

fj = g1 j h1 j g1 j h1 j g1 j h1

79

with a rapidly growing sequence of integer exponents
m1 << n1 << k1 << l1 << s1 << t1 <<
m2 << n2 << k2 << l2 << s2 << t2 <<
m3 << n3 << · · ·
which are chosen by the following protocol. There exists a sequence of functions
M1 ,
M2 ,
M3 ,

N1 , K1 ,
N2 , K2 ,
N3 , · · ·

L1 ,
L2 ,

S1 ,
S2 ,

T1 ,
T2 ,

with the following properties:
• The 1st function M1 has 0 arguments (and so is a constant), the 2nd function
N1 has 1 argument, the 3rd has 2 arguments, and so on, the pth having p − 1
arguments.
• If m1 ≥ M1 , n1 ≥ N1 (m1 ), k1 ≥ K1 (m1 , n1 ), l1 ≥ L1 (m1 , n1 , k1 ), and so on, the
pth exponent being bounded below by the pth function applied to the previous
p − 1 exponents, then properties (4), (5) hold.
At first these functions are chosen to depend on κ ≤ K1 , obtaining a sequence of
functions denoted M1κ , N1κ , . . . , M2κ , N2κ , . . . But then, by the maximization trick, one
replaces the collection of constants M1κ with the single constant M1 = maxκ≤K1 {M1κ };
and one replaces the collection of 1-variable functions N1κ (m1 ) with the single 1variable function N1 (m1 ) = maxκ≤K1 {N1κ (m1 )}; and so on.
First we prove (5). The quasi-isometric embedding ξ1 : T → X restricts to a
ξ1
i
map AT (fi ) −
→ A1 (fi ) taking an h−l
subsegment of AT (fi ) to a subsegment α of
1
A1 (fi ) (see the diagram below). The constant li is chosen so large that if fi ∼ iκ (fj )
then some translate of α stays close rel endpoints to a subsegment of Aκ (fj ) which
ξκ

contains the image under T −
→ X of an entire fundamental domain for AT (fj ) labelled
−sj −tj mj nj kj −lj
m
g1 h1 g1 h1 g1 h1 . Under that translate, the g1 j subsegment of this fundamental
n
domain is taken by ξκ to a subsegment β of Aκ (fj ), and the h1 j subsegment is taken
by ξκ to a subsegment γ of Aκ (fj ), as shown in the the following diagram. In this
diagram the symbol 7→κ is an abbreviation for the map ξκ :
(li repetitions) 7→1 α.

}|
{
z
−1 −1 −1 −1 −1 −1
−1 −1 −1 −1 −1 −1 −si −ti mi ni ki
i mi ni ki
h
h
h
h
h
h
·
·
·
·
·
·
h
h
h
h
h
h
· · · g1−si h−t
g
h
g
1
1
1
1
1
1
1
1
1
1
1
1 g1 h1 g1 h1 g1 · · ·
1
1
1 1
−sj −tj
kj −lj
· · · g1 h1 g1 g1 · · · g1 g1 h1 h1 · · · h1 h1 g1 h1 · · ·
{z
}
|
{z
}|
(mj reps.) 7→κ β.

(nj reps.) 7→κ γ

It follows that some translate of a subsegment of α stays close rel endpoints to βγ,
which implies in turn that there exist integers ξ, η > 0 such that some translate of β
80

stays close rel endpoints to a subsegment of α labelled by h−ξ
1 , and some translate of
γ stays close rel endpoints to a subsegment of α labelled by h−η
1 . Since mj << nj it
follows that ξ < η, and therefore some translate of β stays close rel endpoints to a
subsegment of γ. By taking mj sufficiently large this contradicts g1 6∼κ h1 , item (1)
above. Similarly if fi ∼ iκ (fj−1 ), equivalently if fi−1 ∼ iκ (fj ), then after inverting the
first line in the above diagram, obtaining the segment α−1 labelled by li repetitions of
h1 , one sees that some translate of a subsegment α−1 stays close rel endpoints to βγ,
which leads similarly to a contradiction of g1 6∼κ h1 .
Next we turn to the proof of (4). The quasi-axis A1 (fi ) can be subdivided into
alternating subsegments labelled α1 and β1 which are the images under the map
ξ1
mi ni ki −li −si
i
AT (fi ) −
→ A1 (fi ) of alternating subsegments of AT (fi ) labelled h−t
1 and g1 h1 g1 h1 g1 .
Similarly, Aκ (fi−1 ) can be subdivided into alternating subsegments labelled ακ and
ξκ

→ Aκ (fi−1 ) of alternating subsegments of AT (fi−1 )
βκ , the images under AT (fi−1 ) −
labelled ht1i and g1si hl1i g1−ki h1−ni g1−mi . Note that each α1 β1 subsegment of A1 (fi ) is a
fundamental domain, and similarly for each ακ βκ subsegment of Aκ (fi ). This situation is depicted in the following diagram:
α

A1 (fi ) :
Aκ (fi−1 ) :

β1

α

β1

α

1
1
1
}|
{ z}|{
}|
{ z}|{
z
z
z}|{
−ti mi ni ki −li −si −ti mi ni ki −li −si −ti
· · · h1 g1 h1 g1 h1 g1 h1 g1 h1 g1 h1 g1 h1 · · ·

· · · ht1i g1si hl1i g1−ki h1−ni g1−mi ht1i g1si hl1i g1−ki h1−ni g1−mi ht1i · · ·
|{z} |
{z
} |{z} |
{z
} |{z}
ακ

ακ

βκ

βκ

ακ

For sufficiently large choice of ti >> si , li , ki , mi , ni it follows that α1 is much longer
than βκ and ακ is much longer than β1 .
Arguing by contradiction, suppose that fi ∼ iκ (fi−1 ) for arbitrarily large choices
of ti . Thus, there are subsegments of A1 (fi ) and of Aκ (fi−1 ), each containing arbitrarily many fundamental domains, that are close rel endpoints to each other. This
is depicted in the above diagram, but the alignment of the fundamental domains is
not yet clear; by studying various cases of that alignment we shall arrive in each case
at a contradiction. First, it immediately follows that h1 ∼ iκ (h−1
1 ), because by using
that α1 is much longer than βκ and that ακ is much longer than β1 it follows that
some subsegment of α1 labelled by a large power of h−1
1 is close after translation to
some subsegment of ακ labelled by a large power of h1 .
We now consider four very similar cases of alignment.
−m /2
Case 1: The terminal g1 i subsegment of βκ cannot be close rel endpoints to
a subsegment of α1 : otherwise, since h−1
1 ∼ iκ (h1 ) it would follow that the terminal
−mi /2
g1
segment of βk is after translation close rel endpoints to a subsegment of ακ
labelled by some large power of h1 , which for sufficiently large mi contradicts that
g1−1 6∼κ h1 , item (2).
s /2
Case 2: The initial g1i subsegment of βκ cannot be close rel endpoints to a
subsegment of α1 : otherwise for sufficiently large si we contradict that h1 6∼κ g1 .
81

−s /2

Case 3: The terminal g1 i subsegment of β1 cannot be close rel endpoints to a
subsegment of ακ : otherwise for sufficiently large si we contradict that g1−1 6∼1 h−1
1 .
mi
Case 4: The initial g1 subsegment of β1 cannot be close rel endpoints to a
subsegment of ακ : otherwise for sufficiently large mi we contradict that h−1
1 6∼1 g1 .
As a consequence of Cases 1–4 and the fact that α1 is much longer than βκ and ακ
is much longer than β1 , it follows that α1 and ακ are somewhat close rel endpoints:
the distance between their left endpoints has an upper bound comparable to the
−m /2
−s /2
lengths of the terminal g1 i segment of βκ and the terminal g1 i segment of β1 ,
respectively; and similarly for their right endpoints. But now, using that si >> mi , ni ,
it follows that some long subsegment of the g1−si segment of β1 is close rel endpoints
to the entire h1−ni subsegment of βκ , which for sufficiently large ni contradicts that
g1 6∼1 h1 .
This completes Step 3.
Step 4: Some quasimorphisms on N. We cite Fujiwara [Fuj98] for a method which
associates quasimorphisms to loxodromic elements of hyperbolic group actions. Then
we cite a result of [BF02] giving some cyclic subgroups on which these quasimorphisms
are unbounded and others on which they are bounded. In later steps these results will
be applied to the elements f1 , f2 , . . . constructed in Step 3 and the cyclic subgroups
they generate.
Recall that for each κ ≤ K1 where the action F yκ X is Schottky, and for
each nontrivial g ∈ F , the quasi-axis Aκ (g) is a K, C-quasigeodesic path in the δhyperbolic space X. Using the Morse property of quasigeodesics, we fix a constant
B = B(δ, K, C) such that for any K, C quasigeodesic segment α between points p, q
in a δ-hyperbolic space, the Hausdorff distance between α and any geodesic [p, q] is at
most B. We shall apply this where α is a subsegment of any of the quasi-axes Aκ (g).
Henceforth let W ∈ [3B, 3B +1) be the unique integer, as in [BF02] and in [Fuj98].
Given an edge path η = e1 · · · eK in X let |·| = K denote its length and η =
ēK · · · ē1 its reversal. Two subpaths η ′ = ei′ · · · ej ′ and η ′′ = ei′′ · · · ej ′′ are said to
overlap if {i′ , . . . , j ′ } ∩ {i′′ , . . . , j ′′ } =
6 ∅.
Consider any edge path w in X such that |w| > W . Given another edge path α,
define a copy of w in α to be a subpath of α which is a translate of w by an element
of N, and define
|α|w = the maximal number of nonoverlapping copies of w in α
Then, given vertices x, y ∈ X, define
(∗)

cw (x, y) = d(x, y) − inf (|α| − W |α|w )
α

= sup (W |α|w − (|α| − d(x, y)))
α

where α varies over all edge paths in X from x to y.
82

Remark. If α is a geodesic then the parenthetical quantity inside the supremum of
(∗) is equal to W |α|w which hearkens back to the original method of Brooks in [Bro81]
computing Hb2 for a free group, where X is the Cayley tree of a free group and α is a
geodesic in that tree. When X is a hyperbolic metric space, the method of Fujiwara
in [Fuj98] allows α to travel through a sequence of nonoverlapping “wormholes”—
copies of w—in an attempt to optimize the parenthetical quantities of (∗). With this
allowance a path α which achieves the optimum need no longer be a geodesic, but at
least it is a quasigeodesic (see (6a) below).
Pick once and for all a base point x0 ∈ X. For each edge path w in X define a
function hw : N → R as follows:
(∗∗)

hw (γ) = cw (x0 , γ · x0 ) − cw (x0 , γ · x0 )

(6) Assuming |w| ≥ 2W :
(a) Any edge path α in X which realizes the optimum in (∗) is a quasigeodesic
with constants ℓ = 2, c ≥ 0 depending only on δ, K, C.
(b) For each γ ∈ N we have
|hw (γ)| ≤ H(γ)
where the constant H(γ) is independent of w.
(c) The function hw is a quasimorphism with defect bounded above by a constant D depending only on δ, K, C.
Item (6a) follows from [Fuj98, Lemma 3.3] which says that α is a quasigeodesic with
2W |w|
|w|
≤ 2 = ℓ and additive constant |w|−W
≤ 4W < 12B+4 =
multiplicative constant |w|−W
c. For item (6b), if the edge path α from x0 to γ · x0 realizes the optimum in the first
term cw (x0 , γ · c0 ) of hw (γ) then by (6a) it follows that
0 ≤ cw (x0 , γ · x0 ) ≤ |α| ≤ ℓ d(x0 , γ · x0 ) + c
and the same bound holds for the second term cw (x0 , γ · x0 ). Item (6c) follows from
[Fuj98, Proposition 3.10] which gives a formula for an upper bound to the defect
involving only the quantities δ, W , ℓ = 2, and c = 12B + 4 (the latter two numbers ℓ
|w|
2W |w|
and c being upper bounds for the quantities |w|−W
and |w|−W
as we have just shown).
Next define a quasimorphism h(f ) : N → R for each nontrivial f ∈ F as follows.
First, for each line L ⊂ T pick a base vertex xT (L) ∈ L, chosen so that ξ(xT (L))
is a point of ξ(L) closest to x0 . Next, for each nontrivial f ∈ F define x(f ) =
ξ(xT (AT (f ))) ∈ A(f ), and notice that x(f ) = x(f d ) for all integers d 6= 0. Then
define w(f ) to be an oriented geodesic path in X with initial vertex x(f ) and terminal
vertex f · x(f ). Finally define
h(f ) = hw(f ) : N → R
83

Although it need not be true in general that w(f ) ≥ 2W as needed to apply (6),
as long as f is loxodromic the inequality w(f d ) ≥ 2W is true for sufficiently large
integers d > 0, which leads to:
(7) There is an integer d(f ) > 0 defined for each loxodromic f ∈ N, and an integer
d(f, f ′ ) > 0 defined for each pair f, f ′ ∈ N such that f is loxodromic, such that
the following hold:
(a) If f 6∼ f −1 then for all integers d ≥ d(f ) the values of h(f d ) are nonnegative and unbounded on positive elements of the cyclic group hf i.
(b) If f 6∼ f ′±1 then for all integers d ≥ d(f, f ′) the value of h(f d ) is zero on
each element of the cyclic group hf ′ i.
(c) If f 6∼ f ′−1 then for all integers d ≥ d(f, f ′ ) the values of h(f d ) are nonnegative on positive elements of the cyclic group hf ′ i.
Item (7a), and item (7b) when f ′ is loxodromic, are proved in [BF02] Proposition 5.
Before continuing, here are a few remarks. First we note that [BF02] Proposition 5
has a hypothesis that f be cyclically reduced, which guarantees that if one chooses
x0 ∈ X to be the image under ξ of the identity vertex in the Cayley tree of F then
A(f ) passes through x0 ; but the proof goes through without change using instead that
A(f ) passes through x(f ). We note also that [BF02] Proposition 5 has an implicit
hypothesis w(f d) > W ≥ 3B, which we arrange by requiring d to be sufficiently
large. Finally, we note that item (7b) is a consequence of item (7c) (proved in detail
below) applied once as stated and once with f ′ replaced by f ′−1 .
To prove items (7b) and (7c) when f ′ is not loxodromic, by the WWPD hypothesis
the element f ′ must be elliptic meaning that the diameter of the set {f ′i (x0 )}i∈Z
is bounded by some constant ∆. If d is sufficiently large then w(f d) has length
greater than ℓ∆ + c, which by (6a) is an upper bound for the length of any path that
realizes the optimum for either term of the expression h(f d )(f ′i ) = cw(f d ) (x0 , f ′i (x0 ))−
cw(f d ) (x0 , f ′i (x0 )). It follows that h(f d ) is zero on hf ′ i.
Remark. We do not know how to prove (7b) if one were to allow f ′ to be
parabolic (c.f. the final paragraph of Section 3 of [BF02]).
Here is a proof of item (7c) when f ′ is loxodromic, which is similar to elements
of the proof of [BF02] Proposition 5. If (7c) fails then there exist integer sequences
di → +∞ and ei > 0 such that the following quantity is negative for all i:
h(f di )(f ′ei ) = cw(f di ) (f ′ei ) − cw(f di ) (f ′ei )
It follows that cw(f di ) (f ′ei ) = cw(f di ) (f ′−ei ) is positive for all i. The geodesic path
w(f di ) has the same endpoints as the K, C-quasigeodesic subsegment αi ⊂ A(f ) with
initial endpoint x(f ) and terminal endpoint f di · x(f ), and therefore w(f di ) and αi
are uniformly Hausdorff close rel endpoints. By (6a) there is an ℓ, c quasigeodesic
path βi from x0 to f ′−ei · x0 realizing the optimum in the definition of the number
84

cw(f di ) (f ′−ei ), and since that number is positive it follows that some subsegment of βi
is a translate of w(f di ) and therefore is uniformly Hausdorff close rel endpoints to a
translate of the K, C-quasigeodesic segment αi . Recalling that x(f ′ ) is a point of A(f ′ )
closest to x0 , letting [x0 , x(f ′ )] denote a geodesic segment in X with the indicated
endpoints, and letting γi ⊂ A(f ′−1 ) denote the K, C-quasigeodesic subsegment with
initial point x(f ′ ) and terminal point f ′−ei · x(f ′ ), the following path is also a uniform
quasigeodesic from x0 to f ′−ei ·x0 and so stays uniformly Hausdorff close rel endpoints
to βi :

[x0 , x(f ′ )] ∗ γi ∗ f ′−ei · [x(f ′ ), x0 ]

It follows that some subsegment of the above path is uniformly Hausdorff close rel
endpoints to a translate of αi . Noting that αi is a concatenation of di consecutive
fundamental domains of the axis A(f ), noting also that the prefix [x0 , x(f ′ )] is independent of i, and noting that the suffix f ′−ei [x(f ′ ), x0 ] has length independent of i, it
follows that some subsegment of γi is uniformly Hausdorff close to a translate of a subpath αi′ ⊂ αi consisting of a concatenation of di −d0 consecutive fundamental domains
of A(f ) where d0 is constant. Since di → +∞ it follows that Length(αi′ ) → +∞, from
which it follows that f ∼ f ′−1 , a contradiction.
This completes Step 4.
Step 5: Quasimorphisms on Γ. We now define a sequence of quasimorphisms on Γ
with uniformly bounded defects, denoted
h1 , h2 , h3 , . . . : Γ → R
We shall use this sequence in Step 6 to prove Theorem 2.5.
By combining (4), (5), (6) and (7), we may choose for each i ≥ 1 an integer di ≥ 1
such that the following hold:
(8) The function h(fidi ) : N → R is a quasimorphism with defect bounded by a
constant D ≥ 0 independent of i (by (6c)).
(9) The quasimorphism h(fidi ) is non-negative and unbounded on positive elements
of the cyclic group hfi i (by (4) with κ = 1, and by (7a)).
(10) If 2 ≤ κ ≤ K then the quasimorphism h(fidi ) is non-negative on positive elements of the cyclic group hiκ (fi )i (by (4) with 2 ≤ κ ≤ K, and by (7c)).
(11) If i > j and 1 ≤ κ ≤ K then the quasimorphism h(fidi ) is zero on the cyclic
group hiκ (fj )i (by (5) and (7b)).
Note that the constant di is made independent of κ, and in (11) of j, by the maximization trick.

85

Define functions h′i : N Q ⋊ Sym(Q) → R and hi : Γ → R as follows:
h′i (ρ, φ)

=

hi (µ) =
=
=

K
X

h(fidi )(ρ(Bκ )),

κ=1
h′i (θ(µ))

K
X

κ=1
K
X

for ρ ∈ N Q , φ ∈ Sym(Q)

= h′i (ρµ , φµ ),

for µ ∈ Γ

h(fidi )(ρµ (Bκ ))
h(fidi )(iκ (µ))

in the special case µ ∈ N.

κ=1

Note that hi (µ) has no dependence on φµ , it depends on ρµ alone. We show:
(12) Each hi : Γ → R is a quasimorphism with defect ≤ KD.
For each µ, ν ∈ Γ we have
hi (µ · ν) = h′i (θ(µ · ν)) = h′i (θ(µ) · θ(ν)) = h′i ((ρµ , φµ ) · (ρν , φν ))
= h′i (ρµ · (ρν ◦ φµ ), φν φµ )
K
X

=
KD

h(fidi )(ρµ (Bκ ) · ρν (φµ (Bκ ))) (recall that ρµ (Bκ ), ρµ (φµ (Bκ )) ∈ N)

κ=1
K
X

=

h(fidi )(ρµ (Bκ )) +

κ=1

K
X

h(fiai )(ρν (φµ (Bκ )))

κ=1

where the symbol KD= means that the difference of the two sides has absolute value
bounded by KD; this holds because of (8). Since φµ ∈ Sym(Q) permutes the set of
cosets Q = {B1 , . . . , BK } of N in Γ, by rewriting the second term on the right hand
side of the last equation we obtain:
=

K
X

h(fiai )(ρµ (Bκ )) +

κ=1

= hi (µ) + hi (ν)

K
X

h(fiai )(ρν (Bκ ))

κ=1

This completes Step 5.
Step 6: The proof of Theorem D. To embed
ℓ1 ֒→ Hb2 (Γ; R), consider a sequence of
P
real numbers (t) = (t1 , t2 , t3 , . . .) with
|ti | < ∞. Define the function h(t) : Γ → R
by
h(t) = t1 h1 + t2 h2 + t3 h3 + · · ·
By (12), each hi is a quasimorphism
P with defect ≤ KD, and it follows that h(t) is a
quasimorphism with defect ≤ KD |ti |.
86

The map (t) 7→ h(t) evidently defines a linear map from the vector space ℓ1 to
the vector space of quasimorphisms Γ 7→ R, and so it remains to show that if (t)
is nonzero in ℓ1 then h(t) is unbounded and has unbounded difference with every
homomorphism Γ 7→ R. Letting j ≥ 1 be the least integer such that tj 6= 0 we have:
X
X
ti hi (fjd )
h(t) (fjd ) =
ti hi (fjd ) +tj hj (fjd ) +
i<j

|

{z

i>j

}

=0



K
X X
d

h(fidi )(iκ (fjd ))
= tj
h(fj j )(iκ (fjd )) +
ti 
|
{z
}
κ=1
κ=1
i>j

= tj

K
X

d

h(fj j )(fjd ) +

K
X

d

h(fj j )(iκ (fjd ))

κ=2

!

=0 by (11)

Regarding the quantity in the big parentheses on the last line, as d > 0 varies the
d
summand h(fj j )(fjd ) is non-negative and unbounded by (9), and the other summands
with 2 ≤ κ ≤ K are non-negative by (10), so the whole quantity is unbounded. This
shows that h(t) (fjd ) is unbounded. Also, since fjd ∈ F < [N, N] < [Γ, Γ], the value of
every homomorphism Γ 7→ R on fjd is zero, and so h(t) has unbounded difference with
every such homomorphism.

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