arXiv:1506.04532v2 [] 30 Jan 2017

LONG TURNS, INP’S AND INDICES FOR FREE GROUP
AUTOMORPHISMS
THIERRY COULBOIS AND MARTIN LUSTIG
Abstract. The goal of this paper is to introduce a new tool, called long
turns, which is a useful addition to the train track technology for automorphisms of free groups, in that it allows one to control periodic INPs
in a train track map and hence the index of the induced automorphism.

1. Introduction
Automorphisms of a non-abelian free group FN of finite rank N ≥ 2 are the
focal point of many interesting recent research efforts. The most important
class of such automorphisms are the ones that are irreducible with irreducible
powers (iwip), also called fully irreducible (see §2). Such automorphisms can
always be represented by a map f : Γ → Γ, where Γ is a graph equipped with
∼
=
a marking isomorphism FN −→ π1 (Γ), and f has the train track property:
the map f defines a gate structure G = G(f ) on Γ, i.e. a partition of the edge
germs at every vertex into gates, which is preserved by f in that f maps any
G-legal path to a G-legal image path. Here a path γ in Γ is G-legal if at any
vertex it never enters and exits through the same gate.
In this paper we present a new tool, called long turns, which we use to
define in §5 below legalizing train track morphisms g : Γ → Γ′ (with respect
to fixed gate structures G on Γ and G′ on Γ′ ). A construction device for such
legalizing train track morphisms is given in Section 7.
Theorem 1.1. Let f : Γ → Γ be a train track map which represents an iwip
automorphism of FN and has positive transition matrix M (f ). Let g : Γ → Γ
be a legalizing train track morphism with respect to the gate structure G(f ),
and assume that g induces an automorphism on π1 (Γ) and is gate-stable (see
Definition 3.11). Then:
(1) The map f ◦ g : Γ → Γ is a train track representative of an iwip
automorphism ϕ ∈ Out(FN ).
1991 Mathematics Subject Classification. 20E05, 20E08, 20F65, 57R30.
Key words and phrases. Free group automorphisms, Train tracks, Nielsen path, Index.
1

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THIERRY COULBOIS AND MARTIN LUSTIG

(2) There is no periodic INP in Γ for the train track map f ◦ g. In
particular there are no non-trivial (f ◦ g)-periodic conjugacy classes
in FN .
(3) The stable index list for ϕ is given by the gate index list for f .
To explain the last statement of this theorem we recall from [9] and [18]
that for every automorphism Φ ∈ Aut(FN ) with no periodic w ∈ FN r {1}
the induced action of a suitable positive power of Φ on ∂FN is particularly
simple: There are finitely many attractors and finitely many repellers on ∂FN ,
and every other orbit accumulates positively onto one of the attractors and
negatively onto one of the repellers. The index of Φ, as defined in [9], is in
this special case equal to
a(Φ)
−1,
ind(Φ) :=
2
where a(Φ) denotes the number of attractors of the Φ-action on ∂FN .
For any given ϕ ∈ Out(FN ) it has been shown in [9] that up to isogredience
(= conjugation by inner automorphisms), there are only finitely many lifts
Φk ∈ Aut(FN ) of ϕ which satisfy ind(Φk ) > 0. Those indices form the
index list of ϕ (which is defined up to permutation and typically given in
decreasing order). Through replacing ϕ by a suitable positive power ϕt the
number of terms and also their values in the index list may increase, but
eventually it becomes stable (with respect to passing to further powers, see
[9] or Section 8 below): this is called the stable index list of ϕ. It is an
important invariant of the conjugacy class of ϕ in Out(FN ) with interesting
structural consequences (see [5]). The main result of [9] states that for the
stable index of any ϕ ∈ Out(FN ), i.e. the sum of the values ind(Φk ) in the
stable index list of ϕ, one has a uniform upper bound:
X
(1.1)
indstab (ϕ) :=
ind(Φk ) ≤ N − 1
On the other hand, the gate index list for any train track map f : Γ → Γ
is the list of the gate indices
g(vk )
−1
indG(f ) (vk ) =
2
at the essential vertices vk of Γ, where g(vk ) is the number of gates at vk , and
vk is essential if it has at least 3 gates and is f -periodic. The advantage of
the the gate index list over the above described index list is that it can be
read off directly from the given train track map.
It turns out that Statement 3 of Theorem 1.1 is a consequence of Statements 1 and 2. This follows from standard results on Outer space and R-trees.
For the convenience of the reader we recall and assemble the relevant facts in
Sections 8.
It is a direct consequence of Nielsen-Thurston theory for surface homeomorphisms that in the special case where ϕ is geometric, i.e. ϕ is induced

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by a homeomorphism of some surface with boundary, the above stable index
inequality (1.1) becomes an equality. In general, however, computer experiments of the first author (see [6, subsection 7.3], and [4]) indicate that random
automorphisms have very low stable index; indeed, for up to rank N ≤ 9 more
than half of the investigated automorphisms have index list equal to [ 21 ], to
[ 21 , 12 ], to [1] or to [ 21 , 12 , 12 ].
In our subsequent work [6] we use Theorem 1.1 as crucial tool to produce
explicit automorphisms ϕI ∈ Out(FN ) which realize as stable index list any
given list I = [j1 , . . . , jℓ ] of positive values jk ∈ 21 Z which satisfies the above
inequality (1.1), thus answering a question posed by Handel and Mosher [11].
Theorem 1.1 is also used to derive in Corollary 6.7 information about the
elements of the monoid generated by train track morphisms on a common
graph with respect to a fixed gate structure. This result is also the “door
opener” to a further study of strata in Outer space, in analogy to the well
known and heavily investigated strata in Teichmüller space, see for instance
[22].
Acknowledgements: This paper came into existence largely due to Catherine Pfaff’s postdoc stay in Marseille: It is part of a larger project on which
we started working with her during the last months of her position in Marseille. Although the content of our discussions mostly concern the companion
paper [6] (which was indeed meant to be joint work with her), some of the
material presented in this paper as well must surely be influenced by having
talked to Catherine.
We would also like to thank Ilya Kapovich for some valuable comments on
indices of random automorphisms, in the context of his recent work [16] with
Pfaff.
2. Notation and conventions
Throughout this paper, FN will denote the non-abelian free group of finite
rank N ≥ 2, and Out(FN ) the group of its outer automorphisms. Furthermore, we will use the following conventions and notations:
A graph is always connected, without vertices of valence 1, and finite, unless
e of a finite connected graph Γ. For every oriented
it is the universal covering Γ
edge e of Γ we denote by e the edge with reversed orientation. The set E ± (Γ)
denotes the set of all edges e including their “inverses” e, while by E + (Γ)
we mean a section (sometimes called an orientation) of the quotient map
e 7→ {e, e} on E ± (Γ).
An edge path γ in Γ is a (possibly infinite) sequence γ = . . . ei−1 ei ei+1 . . .
of edges ei ∈ E ± (Γ) where the initial vertex of any ei must agree with the
terminal vertex of ei+1 . Such an edge path γ is reduced if it doesn’t contain any
backtracking subpath γ ′ , i.e. γ ′ is a finite sub-edge-path which has coinciding
initial and terminal vertex, and which is contractible relative to its endpoints.

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THIERRY COULBOIS AND MARTIN LUSTIG

If a finite edge path γ is not reduced, then it is always homotopic rel. endpoints
in Γ to a reduced edge path [γ], and this homotopy can be expressed as
iterative contraction of backtracking subpaths.
The combinatorial length |γ| of a finite edge path γ is equal to the number
of edges traversed by γ, and γ is trivial if |γ| = 0.
A graph map is a map f : Γ → Γ′ between graphs which maps vertices to
vertices and edges e to (not necessarily reduced) edge paths f (e). If f has no
contracted edges, i.e. none of the image paths f (e) is trivial, then f induces a
differential Df : E ± (Γ) → E ± (Γ′ ) which maps e to the initial edge of f (e).
Two edges e and e′ with same initial vertex v form a turn (e, e′ ) at v, and
the map Df induces a map D2 f : (e, e′ ) 7→ (Df (e), Df (e′ )) from the turns of
Γ to the turns of Γ′ . The turn (e, e′ ) is degenerate if we have e = e′ in E ± (Γ).
We say that an edge path γ = . . . ei−1 ei ei+1 . . . crosses over (or uses) a
turn (e, e′ ) if for some index i one has ei = e and ei+1 = e′ .
To every graph map f : Γ → Γ′ there is associated a non-negative transition
matrix M (f ) = (me′ ,e )e′ ∈E + (Γ′ ), e∈E + (Γ) , where the coefficient me′ ,e is the
number of times that the (possibly unreduced) edge path f (e) crosses over the
edge e′ or the inversely oriented edge e′ (both occurrences counted positively!).
It follows directly from the definition that for any two graph maps f : Γ → Γ′
and g : Γ′ → Γ′′ one has:
M (g ◦ f ) = M (g)M (f ).
Recall that a non-negative square matrix M is primitive if there is a positive
power M t with positive coefficients only.
An automorphism ϕ ∈ Out(FN ) is called irreducible with irreducible powers
(iwip) or fully irreducible if there is no proper free factor of FN which is fixed
up to conjugacy by any positive power of ϕ.
A graph map f : Γ → Γ is called expanding if for every edge e ∈ E ± (Γ)
there is an exponent t ≥ 1 such that |f t (e)| ≥ 2. If f represents an iwip
automorphism, then, up to passing to a quotient graph through contracting
f -invariant subtrees in Γ, the map f must be expanding.
3. Gate structures for graphs
Definition 3.1. (1) A gate structure Gv at a vertex v of a graph Γ is a
partition of the edges e ∈ E ± (Γ) with initial vertex v into equivalence classes,
called gates.
(2) A gate structure G on Γ is the collection of gate structures Gv for every
vertex v of Γ. A graph Γ together with a gate structure G on Γ has been
termed a train track in [20].
(3) Two edges e, e′ ∈ E ± (Γ) with same initial vertex v form a legal turn (e, e′ )
(with respect to G) if e and e′ belong to distinct gates. Otherwise the turn
(e, e′ ) is called illegal.

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(4) A (finite or infinite) edge path γ = . . . ei−1 ei ei+1 . . . in Γ is called legal
(with respect to G), if all of the turns (ei , ei+1 ) over which γ crosses are legal.
These notions, introduced in [20] and in a similar fashion elsewhere ([1],
[10], [11], ...), have been inspired by the fundamental paper [2], where the
gate structure G = G(f ) is defined through a train track map f : Γ → Γ, as
explained below in Definition 3.8 and Remark 3.10. There are other natural
occurrences of gate structures, for example given by an edge-isometric FN e to an R-tree T , see [20].
equivariant map from the universal cover Γ

Definition 3.2. Let Γ and Γ′ be graphs equipped with gate structures G and
G′ respectively. A graph map f : Γ → Γ′ is called a train track morphism if
the following two conditions hold:
(1) f has no contracted edges.
(2) f has the train track property: it maps legal paths to legal paths.
Remark 3.3. The reader can verify directly that condition (2) of Definition 3.2 is equivalent to the following two more local conditions:
(2’a) the map D2 f (see Section 2) maps legal turns to legal turns, and
(2’b) for every edge e of Γ the edge path f (e) is legal.

Remark 3.4. (a) It follows directly from Definition 3.2 that the composition
g ◦ f of two train track morphisms f : Γ → Γ′ and g : Γ′ → Γ′′ , with respect
to the same gate structure on Γ′ , is again a train track morphism.
(b) In particular, in the special case Γ′ = Γ and G′ = G we note that for any
edge e and any integer t ≥ 0 the edge path f t (e) is legal and hence reduced.
Remark 3.5. If Γ is equipped with two gate structures G1 and G2 , such
that every gate of G2 is contained in a gate of G1 , then G2 is finer than (or
a refinement of) G1 , while G1 is coarser than G2 .
Any train track morphism f : Γ → Γ′ stays a train track morphism if the
gate structure of Γ is replaced by a coarser one, or the gate structure of Γ′ by
a finer one.
In order to pinpoint certain subtleties which will trouble us later, we define:
Definition 3.6. (a) Let Γ and Γ′ be graphs equipped with gate structures
G and G′ respectively. A graph map f : Γ → Γ′ is called a gate structure
morphism if the following two conditions hold:
(1) f has no contracted edges.
(2) The induced map Df (see Section 2) maps all edges in any gate of
G to a common image gate of G′ . In other words: f induces a well
defined map fG : G → G′ .
(b) If f is not necessarily a gate structure morphism on all of Γ, but induces
a well defined map fGv : Gv → G′f (v) on the gates at a given vertex v of Γ,
we say that f is a local gate structure morphism at v.

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THIERRY COULBOIS AND MARTIN LUSTIG

We’d like to alert the reader that the notions of “train track morphisms”
and “gate structure morphisms” are sort of perpendicular to each other, as
neither of them implies the other: A gate structure morphism is a train track
morphism f if and only if the induced map fG is injective and the image of
every edge is legal. On the other hand, a general train track morphism may
be quite far from being a gate structure morphism, as the identity map which
passes from given gates structure to a strict refinement (compare Remark 3.5)
shows. However, for the special case of graph self-maps we have the following
observation which will turn out to be often quite useful:
Lemma 3.7. Let the graph self-map f : Γ → Γ be a train track morphism
with respect to some gate structure G on Γ.
Then at every f -periodic vertex v of Γ the induced map D2 f maps legal
to legal and illegal to illegal turns. In particular, f induces a well defined
bijective map from the gates at v to the gates at f (v). In other words:
For every periodic vertex v the map f induces a local gate structure morphism fGv : Gv → Gf (v) which is bijective.
Proof. Since f has the train track property, the induced map Df maps edges
in distinct gates to edges in distinct gates. Thus for every vertex v of Γ the
number of gates at f (v) must be larger or equal to the number of gates at v.
Furthermore, if the two numbers are equal, it follows directly that edges in
any given gate must be mapped by Df to edges that also lie all in a common
gate. This implies directly the claimed statement for periodic vertices of a
train track self-morphism.

The most important special case of graph maps, and also the source of
the notion of “gates”, is the case of a self-map f : Γ → Γ. It turns out that
self-maps which don’t even have the train track property define already a gate
structure on Γ:
Definition 3.8. Let f : Γ → Γ be a graph self-map with no contracted
edges. The intrinsic gate structure G = G(f ) is defined by declaring edges
e, e′ with same initial vertex to belong to the same gate if and only if some
power of f maps e and e′ to edge paths which have non-trivial initial subpaths
in common.
Remark 3.9. (1) This definition has built in that f preserves the gates of
the gate structure G(f ):
(a) edges in distinct gates at a common vertex are mapped into distinct
gates and,
(b) all edges in a given gate are mapped into the same gate.
In other words: The graph self-map f is a gate structure morphism with
respect to G(f ), and the induced map fG(f ) is injective.

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(2) Hence, by Remark 3.3, in order to check whether f is a train track morphism with respect to G(f ), we only need to check that for every edge e the
edge path f (e) is legal.
Remark 3.10. (1) Recall that a classical train track map as introduced by
Bestvina and Handel [2] is a graph self-map f : Γ → Γ which has the property
that for any edge e and any integer t ≥ 1 the edge path f t (e) is reduced.
We have already noted in Remark 3.4 (b) that any train track self-morphism
f : Γ → Γ, with respect to any gate structure G on Γ, is such a classical train
track map.
Conversely, it follows directly from the above definitions that every classical
train track map is a train track morphism with respect to the intrinsic gate
structure G(f ).
(2) However, the reader should be warned that for any classical train track
map f as above, in addition to G(f ) there may well be other gate structures
G, with respect to which f is also a train track morphism: For example every
positive automorphism is represented by a train track morphism on the rose
with respect to the gate structure at the sole vertex which consists only of the
“positive” and of the “negative” gates.
(3) On the other hand, it follows directly from the above definitions that
any other such gate structure G must be coarser than G(f ), so that G(f ) is
indeed the finest gate structure with respect to which the train track map f
is a train track morphism.
The following turns out to be a useful notion for the sequel. In order to
properly state it we recall from Lemma 3.7 that any train track self-map f
which fixes a vertex v induces a well defined bijection fGv on the set of gates
at that vertex.
Definition 3.11. A train track morphism f : Γ → Γ with respect to some
gate structure G is called gate-stable if f fixes every vertex of Γ, and at every
vertex f fixes also every gate of G.
We notice directly that every train track self-morphism which acts periodically on every vertex possesses a positive power that is gate-stable.
We say that a path γ crosses over a gate turn (gi , gj ) if γ contains the
subpath e · e′ with e ∈ gi and e′ ∈ gj .
Definition 3.12. Let f : Γ → Γ be a train track morphism with respect to
some gate structure G on Γ.
(a) For every vertex v of Γ we define the gate-Whitehead-graph W hvG (f ) to
be the graph with the set Gv of gates at v as vertex set, and with a (nonoriented) edge connecting gi to gj if for some t ≥ 1 and some edge e ∈ E ± (Γ)
the path f t (e) crosses over v entering through gi and leaving through gj .

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THIERRY COULBOIS AND MARTIN LUSTIG

(b) A vertex v in Γ is essential if it is periodic and if there are at least three
gates at v. The gate index at v is defined as
g(v)
−1
2
where g(v) denotes the number of gates at v.
(c) The gate index list for f is the list of gate indices of G(f ) at essential
vertices. We usually order such a list as decreasing sequence of its values.
indG (v) :=

For graph self-maps the Whitehead graphs at the vertices (in various dialects) have been used previously (e.g. see [2, 11, 12, 14, 20]).
Proposition 3.13. Let f and g be train track morphisms of a graph Γ with
respect to the gate structure G := G(f ). If g induces an automorphism of
π1 (Γ) and is gate-stable, then for every vertex v of Γ the graph W hvG (f ) is a
subgraph of both, W hvG (f ◦ g) and W hvG (g ◦ f ).
Proof. We first use the hypothesis G = G(f ) to deduce (see Remark 3.9 (1))
that f is a gate structure morphism and hence induces a map fG : G → G.
2
Hence there is a well defined map DG
f on the gate turns of Γ, given by
2
DG f (gi , gj ) := (fG (gi ), fG (gj )).
For any vertex v of Γ it follows from the definition of the gate-Whiteheadgraph that in W hvG (f ) two “vertices” gi and gj are connected by an edge if
and only if one of the following occurs:
(a) For some edge e ∈ E ± (Γ) the path f (e) crosses over the gate turn
(gi , gj ).
2 t
(b) For some t ≥ 1 the map DG
f maps one of the gate turns crossed over
by some f (e) to the gate turn (gi , gj ).
From the hypothesis that g is a homotopy equivalence and our convention
that graphs don’t have valence 1 vertices we deduce that each edge e appears
in the image g(e′ ) of some edge e′ . It follows that all gate turns (gi , gj ) crossed
over by the path f (e) are also crossed over by the path f ◦ g(e′ ). Since g is
2
2
gate-stable, the maps gG and DG
g are well defined, and DG
g acts as the
identity on the set of gate turns. This implies both, that g ◦ f (e) crosses
over the same gate turns as f (e), and that the above property (b) for f is
equivalent to property (b) for f ◦ g or g ◦ f .
This shows that, for every vertex v of Γ, in the graph W hvG (f ) two “vertices” gi and gj are connected by an edge only if the same “vertices” are also
connected by an edge in W hvG (f ◦ g) and in W hvG (g ◦ f ).

4. In the absence of INPs
The notion of an INP is a classical concept, going back to [2]. Originally,
“INP” was an abbreviation for “indivisible Nielsen path”; however, through
frequent use it has become mainly an acronym, and as such we treat it here:

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Definition 4.1. Let f : Γ → Γ be a train track morphism with respect to
some gate structure G on Γ. A reduced path η in Γ is called a periodic INP
if η = γ −1 ◦ γ ′ , where the two branches γ and γ ′ are non-trvial legal paths,
and f t (η) is homotopic relative endpoints to η, for some t ≥ 1.
The reader should be alerted that the endpoints of a periodic INP η may
well not be vertices of Γ (although this can be readily achieved by subdividing edges and their iterated f -images, which is a finite procedure since the
endpoints of η are by definition f -periodic).
The importance of the gate-Whitehead-graph (see Definition 3.12) for a
train track map is underlined by the following “irreducibility criterion” from
[12]. We quote here only a simplified version which is used below in Section 6:
Proposition 4.2 ([12, Proposition 5.1]). Let f : Γ → Γ be a train track
representative of ϕ ∈ Out(FN ). Assume furthermore:
(1) The transition matrix M (f ) is primitive.
(2) The gate-Whitehead-graph W hvG(f ) (f ) for f at every vertex v of Γ is
connected.
(3) There is no periodic INP for f in Γ.
Then ϕ is iwip (= fully irreducible).



A “partial converse” of this result is given by the following:
Proposition 4.3 ([12, Proposition 5.1]). Let f : Γ → Γ be a train track
representative of some iwip automorphism ϕ ∈ Out(FN ). Assume that f is
expanding and that there is no periodic INP for f .
Then f must satisfy conditions (1) and (2) from Proposition 4.2.

Remark 4.4. Since the connectedness of the gate-Whitehead-graph for f
at every vertex is a direct consequence of the connectedness of the classical
Whitehead graph of f at v, one can obtain Proposition 4.3 also as direct
consequence of the classical known irreducibility criterion (see [13]).
Alternatively, the reader may prefer to go for a direct proof of Proposition 4.3 according to the following lines: If M (f ) is not primitive, then the
transition matrix M (f t ) of a positive power of f must be reducible, so that
there is an f t -invariant subgraph of Γ, which by expansiveness of f must have
as fundamental group a proper free factor of FN .
Similarly, if W hvG(f ) (f ) is not connected for some vertex v of Γ, then we
can introduce “invisible edges” to blow up that vertex and thus again find,
as complement of the invisible edges, an invariant subgraph with a proper
free factor of FN as fundamental group. (More details regarding the blow-up
technology of vertices by means of invisible edges can be found in the proof
of [12, Proposition 5.1], or in the description before Lemma 6.6 of [14].)

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THIERRY COULBOIS AND MARTIN LUSTIG

5. Long turns
Recall that every graph map f : Γ → Γ′ which induces an isomorphism
on π1 Γ (i.e. f is a homotopy equivalence) possesses a cancellation bound
which can be expressed topologically as a bounded backtracking constant of
e→Γ
e ′ to the universal coverings (see [9]): There exists a constant
any lift fe : Γ
e the unreduced image
C(f ) ≥ 0 such that for any reduced edge path γ in Γ
path fe(γ) is contained in the C(f )-neighborhood of the reduced path [fe(γ)],
e ′ defined by combinatorial path length (see §2). It is
for the metric on Γ
known (see [3, 9]) for homotopy equivalences f that the combinatorial volume
of f (Γ), i.e. the total length of the edge paths f (ei ) for all ei ∈ E+ (Γ), is such
a cancellation bound.
Definition 5.1. Let Γ and Γ′ be graphs equipped with gate structures, and
let f : Γ → Γ′ be a train track morphism.
(1) A long turn (γ, γ ′ ) in Γ is given by two non-trivial legal edge paths γ and
γ ′ with common initial vertex of Γ and distinct first edges.
(2) A long turn (γ, γ ′ ) is called illegal if the initial edges of γ and γ ′ form
an illegal turn (in the sense of Definition 3.1 (3)). Otherwise (γ, γ ′ ) is called
legal.
(3) A long turn (γ1 , γ1′ ) in Γ′ is called the f -image of the long turn (γ, γ ′ ) in
Γ, denoted by
f LT (γ, γ ′ ) := (γ1 , γ1′ ) ,
if f (γ) = γ0 ◦ γ1 and f (γ ′ ) = γ0′ ◦ γ1′ such that γ0 = γ0′ is the maximal common
initial subpath of f (γ) and f (γ ′ ), and γ1 and γ1′ are non-trivial paths.
A long turn which possesses an f -image is called an f -long turn. Otherwise
we call it f -degenerate.
Note that a turn (e, e′ ) in the classical sense of Definition 3.1 (3) is in
particular a long turn, except in the particular case where it is degenerate, i.e.
e = e′ .
Remark 5.2. We note that the existence of a cancellation bound C(f ) for
a train track morphism f : Γ → Γ′ together with the “no contracted edges”
assumption in Definition 3.2 imply that every long turn (γ, γ ′ ) with |γ|, |γ ′ | ≥
C(f ) + 1 is f -long.
We now want to consider the composition g◦f of two train track morphisms
f : Γ → Γ′ and g : Γ′ → Γ′′ . We observe:
(1) If a long turn (γ1 , γ2 ) in Γ is f -degenerate, then f (γ1 ) is an initial subpath
of f (γ2 ) (or conversely), and hence, since γ1 , γ2 as well as their images are
legal, g ◦ f (γ1 ) is an initial subpath of g ◦ f (γ2) (or conversely), so that (γ1 , γ2 )
is also (g ◦ f )-degenerate.

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(2) Similarly, if (γ1 , γ2 ) is f -long, but its f -image long turn (γ1′ , γ2′ ) is gdegenerate, it follows that (γ1 , γ2 ) is also (g ◦ f )-degenerate.
(3) On the other hand, if (γ1 , γ2 ) is f -long and its f -image long turn (γ1′ , γ2′ )
is g-long, then we see directly that (γ1 , γ2 ) is also (g ◦ f )-long, and that its
(g ◦ f )-image coincides with the g-image of (γ1′ , γ2′ ).
We summarize:
Lemma 5.3. Let f : Γ → Γ′ and g : Γ′ → Γ′′ be two train track morphisms
(with respect to the same gate structure on Γ′ ).
A long turn (γ1 , γ2 ) in Γ is (g ◦ f )-long if and only if (γ1 , γ2 ) is f -long and
its f -image long turn f LT (γ1 , γ2 ) is g-long. In this case one has:
(g ◦ f )LT (γ1 , γ2 ) = g LT (f LT (γ1 , γ2 ))

Remark 5.4. If a long turn (γ, γ ′ ) in Γ is legal, then it is f -long for every
train track morphism f : Γ → Γ′ , and the f -image is again legal (indeed, the
common initial subpath γ0 = γ0′ from part (3) of Definition 5.1 is in this case
trivial).
On the other hand, for an illegal f -long turn the f -image can be either
legal or illegal; both cases do occur.
Remark 5.5. (1) Any long turn (γ1′ , γ2′ ) in Γ is called a subturn of a long
turn (γ1 , γ2 ) if each γi′ is an initial subpath of γi , for i = 1, 2. If both long
turns, (γ1 , γ2 ) and (γ1′ , γ2′ ), are f -long for some train track map f , then their
f -images are either both legal or both illegal. Hence, in order to test legality
of the f -image of any long turn, it suffices to calculate the f -image of the
shortest f -long subturn of the given long turn.
(2) If the length of a long turn (γ1 , γ2 ), defined as min{|γ1 |, |γ2 |}, satisfies
min{|γ1 |, |γ2 |} ≥ C ≥ 0, then we will denote below the subturn (γ1′ , γ2′ ) of
(γ1 , γ2 ) with |γ1′ | = |γ2′ | = C by:
(γ1 , γ2 )†C := (γ1′ , γ2′ )
We also denote the set of all long turns (γ1 , γ2 ) in Γ with branch length |γ1 | =
|γ2 | = C by LTC (Γ).
Definition 5.6. A train track morphism f : Γ → Γ′ is called legalizing if
every illegal f -long turn has legal f -image.
It follows from Remarks 5.2 and 5.5 that a train track map f with cancellation bound C(f ) is legalizing if and only if every long turn built from legal
paths of length C(f ) + 1 has legal f -image. Indeed, it follows from Corollary 6.5 below that many (or even “most”) train track representatives of iwip
automorphisms are legalizing.

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THIERRY COULBOIS AND MARTIN LUSTIG

Remark 5.7. The reader should be warned that in the (frequently occurring)
case that a graph map f : Γ → Γ′ is a train track morphism with respect to
a gate structure G on Γ and simultaneously with respect to a coarser gate
structure G0 , then f may well be legalizing with respect to G but not with
respect to G0 .
We should perhaps point out here that although every legalizing train track
morphism maps every path γ with a single illegal turn to a legal path [f (γ)]
(after reduction!), the same does not at all follow for a path γ with more
than one illegal turn. The only conclusion one can draw is that the number of
illegal turns in the reduced image path [f (γ)] is at most half times the number
of illegal turns in γ plus 1.
This notion of a “legalizing map” is robust and easy to handle, as is illustrated by following:
Proposition 5.8. Let f : Γ1 → Γ2 and g : Γ2 → Γ3 be two train track
morphisms. If either f or g is legalizing, then the composition g ◦ f is a
legalizing train track morphism.
Proof. This follows directly from Lemma 5.3, together with the observation
in Remark 5.4 that for any train track morphism and any legal long turn the
image long turn is always legal.

The following observation will be used crucially in the next section:
Lemma 5.9. (a) Let g : Γ → Γ be a train track morphism with respect to
some gate structure G, and assume that g is both, legalizing and gate-stable.
Then the gate structure G is equal to the intrinsic gate structure of g:
G = G(g)
(b) Moreover, if f : Γ → Γ is another train track morphism with respect to G,
then the intrinsic gate structure G(f ◦ g) of the composition f ◦ g is equal G.
Proof. (a) From the hypothesis that g is legalizing we know that the g-image
of every g-long illegal turn (γ, γ ′ ) is a legal long turn. But since g is gatestable, the initial edges of g(γ) and g(γ ′ ) must lie in the same gate, so that
they cannot form a legal turn. Hence they must belong to the common initial
subpath of g(γ) and g(γ ′ ) and thus indeed be identical.
Since this is true for any illegal turn, all legal paths exiting from the same
gate must have g-images with coinciding initial edge. This proves that the
gate structure G is finer than or equal to G(g). The converse is true for
any self-map g that is a train track morphism with respect to a given gate
structure G, see Remark 3.10 (3).
(b) Since f is a train track morphism and hence has no contracted edges, the
above proved fact, that Dg maps all edges in any given gate to a single edge,

LONG TURNS, INP’S AND INDICES FOR FREE GROUP AUTOMORPHISMS

13

is inherited by D(f ◦ g). Hence the arguments from the previous paragraph
are also true for f ◦ g instead of g, so that we obtain G(f ◦ g) = G.

Remark 5.10. (1) Note that the proof of the last lemma stays valid if the
hypothesis “g gate-stable” is replaced by the weaker assumption ”g gate structure morphism” (see Definition 3.6). In particular, by Lemma 3.7 it suffices
to assume that all vertices of Γ are periodic under the map g.
(2) Note also that there is a delicacy in Statement (b) of the last lemma: The
analogous statement for the composition g ◦ f is in general wrong, unless one
assumes that f is a gate structure morphism.
6. Legalizing maps for iwip automorphisms
We will now concentrate on the situation of a classical train track map
f : Γ → Γ, which is a train track morphism with respect to the intrinsic gate
structure G(f ) on Γ, see Remark 3.10. We assume furthermore that f is a
homotopy equivalence so that it possesses a cancellation bound C(f ), and
that it satisfies the following expansion property:
Definition 6.1. For any constant K ≥ 1 a train track morphism f : Γ → Γ′
is called strongly K-expanding if every legal edge path γ in Γ of length |γ| ≥ K
has f -image which is strictly longer:
|f (γ)| ≥ |γ| + 1.
Remark 6.2. Expanding train track morphisms (see §2) are not necessarily
strongly K-expanding for some K ≥ 1 (and conversely), but it follows directly
from the definitions that every expanding train track morphism has a positive
power which is strongly 1-expanding.
We define the minimal stretching factor λK
min (f ) of f for legal paths of
length ≥ K by:
λK
min (f ) := min{

|f (γ)|
| γ legal of length |γ| ≥ K}
|γ|

We will now derive from any cancellation bound C(f ) ≥ 0 of a strongly
K-expanding train track morphism f : Γ → Γ an expansion bound C(f )+ ≥ 0:
Lemma 6.3. Let f : Γ → Γ be a train track map which possesses a cancellation bound C(f ) and is strongly K-expanding for some K ≥ 1. We define:
C(f )+ := max(K,

C(f )
).
λK
min (f ) − 1

Let C ≥ C(f )+ be an integer. Then the map f induces a map
f LTC : LTC (Γ) → LTC (Γ), (γ, γ ′ ) 7→ f LT (γ, γ ′ )†C .

14

THIERRY COULBOIS AND MARTIN LUSTIG

Proof. From the definition of the minimal stretching factor λK
min (f ) it follows
that every legal path γ of length |γ| ≥ C(f )+ ≥ K is mapped by f to a legal
path of length |f (γ)| ≥ |γ| + C(f ). Hence it follows from the definition of a
cancellation bound C(f ) that any long turn (γ, γ ′ ) ∈ LTC (Γ) is f -long, and
that its image long turn f LT (γ, γ ′ ) has length ≥ C. Thus setting (γ, γ ′ ) 7→
f LTC (γ, γ ′ )†C defines indeed a well defined map f LTC : LTC (Γ) → LTC (Γ).

Proposition 6.4. Let f : Γ → Γ be a train track map which possesses a
cancellation bound C(f ) and is strongly K-expanding for some K ≥ 1. Let
C ≥ C(f )+ .
Then for any f LTC -periodic illegal long turn (γ, γ ′ ) ∈ LTC (Γ) the concatenation γ ◦ γ ′ contains a periodic INP as subpath. Conversely, every periodic
INP in Γ can be prolonged on both sides so that the two legal branches give
an f LTC -periodic illegal long turn in LTC (Γ).
Proof. We know from Lemma 6.3 that f induces a well defined map f LTC on
the long turns in LTC (Γ). Assume now that for some integer t ≥ 1 the long
turn (γ, γ ′ ) ∈ LTC (Γ) is illegal and fixed by (f LTC )t . Then γ is a subpath of
f t (γ), and γ ′ is a subpath of f t (γ ′ ). Thus on both legal paths γ and γ ′ there
must be a fixed point, which by the illegality of the turn must be different
from the initial vertex of both, γ and γ ′ . We can define η to be the path
crossing over the illegal turn and connecting those two fixed points. Then
[f t (η)] = η, and since η crosses over precisely one illegal turn, it follows that
it is a periodic INP.
Conversely, it follows from a standard calculation that the legal branches of
any periodic INP η can not be longer than C(f )+ . Thus they can be prolonged
by legal paths so that this prolongation gives a long turn (γ, γ ′ ) ∈ LTC (Γ).
Let now the integer t ≥ 1 be such that [f t (η)] = η. We consider the iterates
of (γ, γ ′ ) under f t and note that they all contain η as subpath, in such a way
that the illegal turn on η coincides with the illegal turn formed by (γ, γ ′ ) (and
thus also by all of its f t -iterates). Since LTC (Γ) is finite, eventually some such
iterate f kt (γ, γ ′ ) will be (f LTC )t -periodic. This shows the “converse” direction
of the claim.

Corollary 6.5. Let f : Γ → Γ be an expanding train track map that represents
an automorphism of FN . Then precisely one of the following is true:
(a) The map f possesses a periodic INP, or
(b) every sufficiently high power of f is legalizing for G(f ).
Proof. Since f represents an automorphism, it possesses a cancellation bound.
Since f is expanding, any sufficiently large power of f will be strongly Kexpanding for K = 1 (see Remark 6.2). Thus Proposition 6.4 applies, so that,
in case that f does not possess a periodic INP, we can deduce that there is
no illegal f -periodic long turn (γ, γ ′ ) ∈ LTC (Γ), for C as in Proposition 6.4.

LONG TURNS, INP’S AND INDICES FOR FREE GROUP AUTOMORPHISMS

15

It follows that after applying f iteratively at least k0 := card LTC (Γ) times,
any long turn (γ, γ ′ ) ∈ LTC (Γ) must have become legal. Since it stays legal
under further iteration of f (see Remark 5.4), every f k with k ≥ k0 must be
legalizing.
Clearly both (a) and (b) can not hold simultaneously. Thus we have proved
the desired dichotomy.

One can derive from the last proof that the lower bound for the exponent
of f needed in statement (b) of Corollary 6.5 can be efficiently calculated from
the train track map f with not much effort. It turns out that it only depends
on the cancellation bound C(f ) and on the rank N of the free group FN .
Proof of Theorem 1.1. Since g is assumed to be a train track morphism with
respect to the gate structure G(f ), then so must be f ◦ g. Since g induces an
automorphism on π1 Γ, so does f ◦ g, so that f ◦ g is a train track morphism
which represents an automorphism of FN .
Recall from Section 2 that the transition matrix of f ◦ g is obtained as
product M (f ◦ g) = M (f ) · M (g). Hence M (f ◦ g) inherits positivity from the
assumed positivity of M (f ). Thus in particular M (f ◦ g) is primitive.
As g is gate-stable and legalizing for G(f ), by Lemma 5.9 the intrinsic gate
structure G(f ◦ g) is equal to G(f ).
By hypothesis, f represents an iwip automorphism. From Proposition 4.3,
we know that for any vertex v the graph W hvG (f ) is connected. As g is gatestable, by Proposition 3.13 the gate-Whitehead-graph W hvG (f ◦ g) must also
be connected.
We now observe from Proposition 5.8 that the composed map f ◦ g must
be legalizing, which implies by Corollary 6.5 that there are no periodic INPs
for f ◦ g.
Thus the conditions (1), (2) and (3) of Proposition 4.2 are all satisfied for
the map f ◦ g, which hence must induce an iwip automorphism. Corollary 8.5
concludes the proof.

Remark 6.6. (1) We see from the above proof that the hypotheses in Theorem 1.1 can be weakened somewhat: In the proof it is never used that the
automorphism represented by f is iwip. It suffices to assume that M (f ) is
positive, and that the gate-Whitehead-graph at every periodic vertex is connected.
(2) We also don’t use the fact that the gate structure on Γ is equal to G(f ).
It suffices to assume that f and g are train track morphisms with respect to
some fixed gate structure G, if in statement (3) of Theorem 1.1 the list of
gate indices at the f -periodic vertices of Γ is computed with respect to the
gate structure G (see Definition 3.12). This is a consequence of Lemma 5.9.
(3) A careful analysis of the above proof and its various ingredients, shows
that the statement of Theorem 1.1 is valid as well for the map g ◦ f in place

16

THIERRY COULBOIS AND MARTIN LUSTIG

of f ◦ g, if one assumes in addition that f is a gate structure morphism (see
Remark 5.10 (2)).
We’d like to remark here that part (2) of the previous remark gives the
possibility to produce, from a given train track map f with a fine gate structure, through properly choosing the legalizing “perturbation map” g, a variety
of train track maps f ◦ g with coarser gate structures and thus, via part (3)
of Theorem 1.1, with smaller index lists than f . A useful technology for
the deliberate production of such perturbation maps is described in the next
section.
We conclude this section by passing to a larger set of product maps:
Corollary 6.7. Let Γ be a graph equipped with a gate structure G, and for
any index i of some index set I let fi : Γ → Γ be a train track morphism with
respect to G. Assume that each fi satisfies the following properties:
(1) The intrinsic gate structure satisfies G(fi ) = G.
(2) The transition matrix M (fi ) is positive.
(3) For any vertex v of Γ the graph W hvG (fi ) is connected.
(4) There is no periodic INP for fi in Γ.
(5) The map fi is gate stable.
Then there exist exponents mi ≥ 1 such that the properties (2) - (5) hold for
every element in the monoid generated by the fimi , i.e. for any product
mi

mi2

f = fi1 1 fi2

m

· · · fis is

of the fimi (but not their inverses!). Furthermore, any such map f represents
an iwip automorphism, the map f is legalizing, and the index list of f is equal
to the list of gate indices of G at the vertices of Γ with 3 or more gates.
Proof. By Corollary 6.5 there exist exponents mi ≥ 1 such that each of
the maps fimi is legalizing. Furthermore, conditions (2), (3) and (4) ensure
via Proposition 4.2 that fi represents an iwip automorphism of FN . Conditions (2), (3) and (5) are inherited by products, if they are satisfied by
every factor. The same is true for the property “legalizing”, which implies
condition (4). By Lemma 5.9 condition (1) is a consequence of condition (5)
together with the property “legalizing”.
Hence all conditions for the factors f and g in Theorem 1.1 are satisfied
for any of the maps fimi as well as for any product f as above. Thus the
conclusion (3) of Theorem 1.1 hold as well for f , which proves the last assertion
in the statement of Corollary 6.7.

The above corollary admits a natural extension to a more involved situation, where one considers simultaneously several graphs Γk with gate structures Gk , as well as maps fi : Γk → Γk′ which induce bijections on the vertices
with 3 or more gates, as well as bijections on the set of their adjacent gates.
This leads one directly to consider “strata” in Outer space, in analogy to

LONG TURNS, INP’S AND INDICES FOR FREE GROUP AUTOMORPHISMS

17

strata in Teichmüller space as defined by fixing the indices of the singularities
of quadratic differentials, see [22].
7. Legalizing Factory
In this section we reduce the construction of a legalizing train track morphism to the construction of a family of “elementary” train track morphisms
that each legalizes only a single illegal turn.
Proposition 7.1. Let Γ be a graph equipped with a gate structure G. Assume
that there exists an integer L ≥ 1 which satisfies:
(1) For each illegal long turn t = (γ, γ ′ ) of branch length L there exists a
train track morphism gt : Γ → Γ such that t is gt -long and mapped by
gtLT to a legal long turn.
(2) There exists a train track morphism h : Γ → Γ which is strongly
K-expanding for some K ≥ 1.
We assume furthermore that each of the above maps gt and h has a cancellation bound C(gt ) or C(h) respectively (which is true if they induce automorphisms of π1 (Γ)). Then there exists a legalizing train track morphism
g : Γ → Γ which is obtained as a composition of the gt and h.
Proof. For each of the illegal long turns t of G with branch length L set
gt′ := h ◦ gt , and observe that gt′ is strongly K-expanding and inherits a
cancellation bound C(gt′ ) from h and gt . Moreover, as h is a train track
morphism, it maps legal turns to legal turns, so that by hypothesis (1) the
long turn t is gt′ -long and mapped by gt′ to a legal long turn.
Let C be the maximum of L and of all the constants C(gt′ )+ , as defined in
Lemma 6.3 for any of the maps gt′ via the cancellation bounds C(gt′ ) and the
LT
above constant K. Then each gt′ induces a well defined map gt′ C on the set
of long turns LTC (Γ).
We can now build iteratively the legalizing train track morphisms we are
looking for: Let g0 be the identity map and LTCill (g0 ) ⊂ LTC (Γ) be the finite
set of illegal long turns in Γ of branch length C. We define iteratively graph
maps gk : Γ → Γ and nested subsets
LTCill (gk )

LTCill (gk−1 )

...

LTCill (g0 )

by considering any turn t∗ in LTCill (gk ). From the iterative definition of
LTCill (gk ) it follows that t∗ is mapped by gkLTC to long a turn (of branch
length C) which is illegal. Let t be the subturn of gkLTC (t∗ ) of branch length
L ≤ C, which is of course also illegal. We set gk+1 := gt′ ◦ gk , and define
LTCill (gk+1 ) to be the set of illegal long turns in Γ of branch length C that are
LTC
mapped by gk+1
to an illegal long turn. Note that gt′ was defined so that t∗
LT
is mapped by gk+1
to a long turn that contains the legal long turn gt′LT (t) as
subturn and is therefore legal. Recall that if a long turn is mapped by gkLTC

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THIERRY COULBOIS AND MARTIN LUSTIG

to a legal long turn, then, as gt′ is a train track morphism, it is also mapped
LTC
by gk+1
to a legal long turn. In other words LTCill (gk+1 ) LTCill (gk ).
From the finiteness of LTCill (g0 ) we deduce that after finitely many steps
one gets gn with LTCill (gn ) = ∅, which is equivalent to stating that g = gn is
legalizing.

8. Stable indices, branching indices and gate indices
The content of this section is well known to the experts, or in close proximity of well known facts; we assemble them here for the convenience of the
reader. We will use some standard tools from R-trees and Outer space technology. For background and terminology the reader may consult [20]; further
detail can be found in [7] or [23]. We follow here mostly the original source
[9].
For every expanding train track map f : Γ → Γ there exists a non-negative
real eigenvector ~v of the transition matrix M (f ) which has real eigenvalue
λ > 1, and any such ~v determines an R-tree T = T (~v) (which in some cases
is called the forward limit tree and can be considered as boundary point of
Outer space CVN ).
e→Γ
e of
The tree T = T (~v) is obtained by choosing an arbitrary lift fe : Γ
e
the train track map f to the universal covering Γ, and by defining T to be the
e which is the limit for t →
metric space associated to the pseudo-metric d∞ on Γ
∞ of the decreasing sequence of pseudo-metrics dt (x, y) := λ1t d~v (fet (x), fet (y)).
e is defined through lifting the ~v -edge-lengths
Here the pseudo-metric d~v on Γ
of Γ that are explicitly given by the coefficients of the eigenvector ~v .
e→T
As a consequence one obtains a canonical FN -equivariant map i : Γ
which is edge-isometric with respect to the pseudo-metric d~v , i.e. every edge
e is mapped by i isometrically to its image i(e) ⊂ T . The map fe also
e of Γ
induces directly a homothety H : T → T with stretching factor λ, and one
obtains the following “commutative diagram”:
(8.1)

H ◦ i = i ◦ fe

e isometrically to segments in T . On the other
The map i maps legal paths in Γ
e which is the lift of a periodic INP in Γ is folded by i
hand, any path ηe in Γ
completely to a single segment, which is the isometric image of any of the two
legal branches of ηe.
It follows from standard train track arguments (see for example Section 3
of [14]) that for any path γ in Γ a sufficiently high f -iterate f t (γ) is homotopic rel. endpoints to a pseudo-legal path, i.e. a legal concatenation of legal
e one has
paths and periodic INPs. It follows that for any two points x, y ∈ Γ
e
i(x) = i(y) if and only if after iterating f sufficiently many times the geodesic
e which joins fet (x) to fet (y) is a legal concatenation of
path γ
e in the tree Γ

LONG TURNS, INP’S AND INDICES FOR FREE GROUP AUTOMORPHISMS

19

legal subpaths and lifts of periodic INPs, where the legal subpaths only use
edges with d~v -length 0. In particular, we see that the absence of INPs for f
implies directly that the FN -action on T is free, if all the exponents of the
the eigenvector ~v are positive. The latter is known if the transition matrix
M (f ) is primitive, and hence always true if the expanding train track map
f represents an iwip automorphism. Furthermore, the North-South result of
the ϕ-action on the closure of CVN proved in [17] yields:
Proposition 8.1. For any iwip automorphism ϕ the forward limit tree T =
T (~v) is well defined up to uniform rescaling of the metric, and in particular
does not depend on the expanding train track representative f : Γ → Γ and its
primitive transition matrix M (f ) with Perron-Frobenius eigenvector ~v .
If Γ doesn’t contain any non-trivial loop which is a legal concatenation of
periodic INPs, then the FN -action on T by isometries is free. This conclusion
is in particular true if there is no periodic INP in Γ.

e
As a direct consequence of the above described construction of T from Γ
by means of the eigenvector ~v one has the following fact, which is well known
to the experts (see [9], [11], [20], or, for much detail, § 7 of [15]). Recall that
a direction at a point P ∈ T is a connected component of T r {P }.
Proposition 8.2. Let f : Γ → Γ be an expanding train track map, and let
~v be an eigenvector of M (f ) with eigenvalue λ > 1. Assume that ~v has only
positive coefficients, and let T = T (~v) the corresponding forward limit tree.
(1) If there is no periodic INP for f , then the map i restricts to an FN e (i.e. lifts of f equivariant bijection iV between essential vertices vk of Γ
periodic vertices of Γ with 3 or more gates) on one hand, and branch points
i(vk ) of T on the other. This bijection extends to a canonical bijection between
the gates at any vk and the directions at i(vk ) (where a gate gj is mapped to
◦
the direction that contains the open segments i(ei ) for any edge ei in gj ).
(2) [Not used in the sequel.] If f possesses periodic INPs, then the endpoints
(assumed to have been made into vertices) of any such periodic INP η have to
be considered as equivalent, and for these endpoints one has to identify those
two gates which contain the two branches of η. Then we get the precisely
analogous statement as in the “no INP” case (1), except that the preimage of
a branch point in T will now be the lift of an f -periodic equivalence class of
vertices in Γ with (after the above identification) 3 or more gates.

If X is a topological space, provided with a left action of a group G by
homeomorphisms, we say that a map F : X → X represents an automorphism
Φ ∈ Aut(G) if for all x ∈ X and all g ∈ G one has:
Φ(g) · F (x) = F (g · x)
This applies in particular to the special case where X is the universal covering
of the quotient space X/G with π1 (X) = G, and F induces a homeomorphism

20

THIERRY COULBOIS AND MARTIN LUSTIG

f : X/G → X/G. In this case, if F represents Φ, then f induces on π1 (X) = G
the outer automorphism ϕ defined by Φ.
It follows from the equality (8.1) and the FN -equivariance of the map i
that any lift fe of the train track map f represents the same automorphism
Φ ∈ Aut(FN ) as the associated homothety H : T → T , where Φ induces (by a
the previous paragraph) the outer automorphims ϕ that is represented by the
train track map f . Since the stretching factor of H satisfies λ > 1, it follows
that H has precisely one fixed point Q which is either contained in T , or it
lies in the metric completion T of T (where we use the canonical extension of
H to T ).
Let now Φ′ ∈ Aut(FN ) be a second lift of ϕ, and assume that Φ′ is isogredient to Φ, i.e. Φ′ = ιw ◦ Φ ◦ ιw−1 = ιw ◦ ιΦ(w)−1 ◦ Φ for some w ∈ FN , where
ιv : FN → FN denotes the conjugation u 7→ vuv −1 . Assume furthermore that
e →Γ
e of f and the homothety H ′ : T → T both represent the
the lift fe′ : Γ
′
automorphism Φ . Then we obtain fe′ = wfew−1 and H ′ = wHw−1 , and thus
deduce for the fixed point Q′ of H ′ the equality Q′ = wQ.
Conversely, if H ′ = uH is the homothety of T which represent some lift
′
Φ = iu Φ of ϕ, then, if H ′ (Q) = Q, the action of u on T must fix the point
Q. Thus, if the FN -action on T is free, then one deduces u = 1.
As a consequence one gets a natural injective map from the isogredience
classes of lifts Φk of ϕ into the set of FN -orbits of points in T , given by the
fixed point Qk of the associated homothety Hk : T → T that represents Φk .
In [9, Theorem 2.1 (3) and Proposition 4.4], the following has been proved
(we only cite the easy case where Stab(Q) is trivial):
Proposition 8.3. Let ϕ ∈ Out(FN ) be an iwip automorphism, and let T =
T (~v) be the forward limit tree of ϕ, given as above by some eigenvector ~v with
eigenvalue λ > 1 of the transition matrix M (f ) of a train track representative
f : Γ → Γ of ϕ. We assume that there is no periodic INP in Γ, so that the
FN -action on T is free.
Let H : T → T be the homothety (with stretching factor λ) that represents
some lift Φ ∈ Aut(FN ) of ϕ, and assume Φ has index Ind(Φ) > 0. Then the
fixed point Q of H is contained in T , and Q is a branch point of T . There is
a natural injection iQ from the set of attracting fixed points of Φ on ∂FN to
the set of directions at Q ∈ T . The image of iQ is precisely the set of those
directions that are fixed by H.

From the last sentence of this proposition we see that replacing ϕ (and
hence also Φ and H) by a positive power will increase the image set of the
map iQ . From Gaboriau and Levitt [8] finiteness result, one knows that for any
T with free FN -action every branch point has only finitely many directions,
so that a suitable positive power of H will indeed fix every direction of T at
Q.

LONG TURNS, INP’S AND INDICES FOR FREE GROUP AUTOMORPHISMS

21

The same finiteness result [8] also implies that there are only finitely many
FN -orbits of branch points in T . Thus through possibly replacing ϕ by a
further positive power we can assume that the associated homothety H of T
fixes every FN -orbit of branch points of T . Thus for any branch point Q′ of
T there is a suitable element u ∈ FN such that the homothety H ′ = uH has
Q′ as fixed point (pick u such that H(Q′ ) = u−1 Q′ ). Now we “perturb” Φ
and fe correspondingly to obtain Φ′ = ιu ◦ Φ and fe′ = ufe, and, if need be, we
pass to another common positive power, so that the homothety H ′ fixes every
direction of T at Q′ . As a consequence, the map iQ′ from Proposition 8.3
becomes a bijection between the attracting fixed points of Φ′ on ∂FN and the
directions of T at Q′ .
Thus we obtain:
Proposition 8.4. Let ϕ ∈ Out(FN ) be an iwip automorphism that has an
expanding train track representative f : Γ → Γ without periodic INPs, and let
T be its forward limit tree.
Then, for some integer t ≥ 1, there exists a natural bijection between on
one hand the isogredience classes of representatives Φk of ϕt which satisfy
ind Φk ≥ 21 , and on the other hand the FN -orbits FN · Qk of branch points Qk
of T .
This correspondence extends further to a bijection between the set of attractors for the induced Φk -action on ∂FN , and the set of directions of T at
Qk .
The assertion remains valid if the integer t is replaced by any positive integer t′ = kt ∈ N.

Recall from the Introduction that the stable index list of an automorphism ϕ ∈ Out(FN ) without non-trivial periodic conjugacy classes is given
by the maximal (decreasing) list of indices ind(Φk ) = a(Φ2 k ) − 1 for representatives Φk ∈ Aut(FN ) of a suitable positive power of ϕ that are pairwise
non-isogredient, where a(Φk ) denotes the number of attractors of Φk on ∂FN .
Thus Proposition 8.4 shows that for an iwip automorphism ϕ, assumed
to have an expanding train track representative f : Γ → Γ without periodic
INPs, the stable index list of ϕ agrees with the branching index list of T ,
i.e. the maximal (decreasing) sequence of values ind(Qk ) := b(Q2 k ) − 1 for
branch points Qk in distinct FN -orbits, where b(Qk ) denotes the number of
directions of T at Qk . On the other hand, we obtain directly from part (1) of
Proposition 8.2 that the branching index list of T agrees with the gate index
list for the map f : Γ → Γ, as defined in Definition 3.12. Thus we obtain:
Corollary 8.5. Let ϕ ∈ Out(FN ) be an iwip automorphism, and let f : Γ → Γ
be an expanding train track map which represents ϕ and which doesn’t have
any periodic INP.

22

THIERRY COULBOIS AND MARTIN LUSTIG

Then the stable index list of the automorphism ϕ agrees with the gate index
list of the map f .

In fact, we see from the details of the above correspondences that, after raising ϕ and f to a suitable common positive power, there is a natural bijection
between the essential vertices of f and the isogredience classes of ϕ, which
for any essential vertex vk and any representative Φk of the corresponding
isogredience class extends further to a bijection between the attracting fixed
points of Φk on ∂FN and the gates at vk . This last bijection can be seen
concretely by considering the unique eigenray ρ defined by any gate at vk , its
lift to an eigenray ρe of a (suitably chosen) lift fek of f for the universal covering
e and the image i(e
Γ,
ρ) ∈ T , which is an eigenray of the associated homothety
Hk which represents Φk . Under the canonical FN -equivariant identification
e the ray ρe represents an attractor of the Φk -action on ∂FN , and
∂FN = ∂ Γ
conversely, every attractor for Φk comes from such an eigenray.
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Institut de mathématiques de Marseille, Université d’Aix-Marseille, 39, rue
Frédéric Joliot Curie, 13453 Marseille Cedex 13, France, thierry.coulbois@
univ-amu.fr, martin.lustig@univ-amu.fr

