FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

arXiv:1610.04098v3 [math.GT] 6 Oct 2017

SANG-HYUN KIM, THOMAS KOBERDA, AND MAHAN MJ
Abstract. We develop a general framework for producing uncountable families
of exotic actions of certain classically studied groups acting on the circle. We
show that if L is a nontrivial limit group then the nonlinear representation variety HompL, Homeo` pS 1 qq contains uncountably many semi-conjugacy classes of
faithful actions on S 1 with pairwise disjoint rotation spectra (except for 0) such
that each representation lifts to R. For the case of most Fuchsian groups L, we
prove further that this flexibility phenomenon occurs even locally, thus complementing a result of K. Mann. We prove that each non-elementary free or surface
group admits an action on S 1 that is never semi-conjugate to any action that factors
through a finite–dimensional connected Lie subgroup in Homeo` pS 1 q. It is exhibited that the mapping class groups of bounded surfaces have non-semi-conjugate
faithful actions on S 1 . In the process of establishing these results, we prove general combination theorems for indiscrete subgroups of PSL2 pRq which apply to
most Fuchsian groups and to all limit groups. We also show Topological Baumslag
Lemma, and general combination theorems for representations into Baire topological groups. The abundance of Z–valued subadditive defect–one quasimorphisms
on these groups would follow as a corollary. We also give a mostly self-contained
reconciliation of the various notions of semi-conjugacy in the extant literature by
showing that they are all equivalent.

Contents
1. Introduction
1.1. Combination Theorems and Indiscrete Subgroups of PSL2 pRq
1.2. Uncountable Families of Exotic Group Actions on the Circle
1.3. An Axiomatic Approach to Combination Theorems
1.4. Flexibility and Rigidity
1.5. Mapping Class Groups
1.6. Notes and References
1.7. Outline of the Paper
2. Preliminary

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Date: October 10, 2017.
2010 Mathematics Subject Classification. 57M05, 37E45 (Primary); 20F65 57S05 (Secondary).
Key words and phrases. Rotation spectrum, limit groups, surface group representation, Baumslag
Lemma, Combination Theorem.
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S. KIM, T. KOBERDA, AND M. MJ

2.1. Actions on the Circle
2.2. Hyperbolic Geometry
2.3. Indiscrete Subgroups of PSL2 pRq
3. Topological Baumslag Lemma
3.1. Topological Setting
3.2. Projective and Discrete Settings
4. Splittable Fuchsian Groups
4.1. Very General Points, Abundance and Stable Injectivity
4.2. Pulling-Apart Lemma
4.3. Almost Faithful Paths
4.4. Simultaneous Control of Rotation Numbers
5. Combination Theorem for Flexible Groups
5.1. Statement of the result
5.2. Proof
5.3. Exotic Circle Actions
5.4. Quasi-Morphisms
5.5. Limit Groups
6. Axiomatics
6.1. Tracial Structure
6.2. UV-structure
6.3. Combination Theorem for Smooth Actions
7. Mapping Class Groups
7.1. The Universal Circle and Nielsen’s Action
7.2. Exotic Mapping Class Group Actions
8. Zero rotation spectrum and Teichmüller theory
8.1. Rigidity of Projective Actions
8.2. Lie Subgroups of the Circle Homeomorphism Group
8.3. Free and surface subgroups of π1 pMq
8.4. Nonlinear Smooth Actions of Free Groups
Appendix A. Equivalent Notions of Semi-Conjugacy
Acknowledgments
References

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1. Introduction
In this paper, we study finitely generated groups which are classically known to
act faithfully on the circle. The purpose of this article is to systematically construct
uncountable families of actions of these groups which have “essentially different”

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

3

dynamics. The tools we develop allow us to construct many exotic actions of classically studied groups, i.e. actions which are not semi-conjugate to the “usual” or
“standard” actions of these groups.
In the course of developing tools to build exotic group actions, we prove several combination theorems, which construct a framework for building dense subgroups of PSL2 pRq, viewed as a subgroup of the group Homeo` pS 1 q of orientation–
preserving homeomorphisms of the circle. Dense subgroups of PSL2 pRq fall outside
the purview of classically studied objects, and to study them we employ methods
from representation varieties, dynamics in one dimension, and hyperbolic geometry
in two and three dimensions.
Throughout this introduction, we will let L be a finitely generated group. By a
circle action of L, we will mean a homomorphism
ρ : L Ñ Homeo` pS 1 q.
A circle action ρ of L is liftable if ρ factors as
L

/

HomeoZ pRq

/

Homeo` pS 1 q ,

where HomeoZ pRq is the set of orientation-preserving homeomorphisms f on the
real line satisfying
f px ` 1q “ f pxq ` 1.
Roughly speaking, we say that two circle actions ρ0 and ρ1 of L are semi-conjugate,
and write ρ0 „semi ρ1 , if there is a monotone degree one map h : S 1 Ñ S 1 such that
h ˝ ρ0 “ ρ1 ˝ h
after “lifting” the actions to R; see Definition 2.2. We say that ρ0 and ρ1 are
equivalent if there exists an α P AutpLq such that ρ0 ˝ α is semi-conjugate to ρ1 .
Semi-conjugacy and equivalence are indeed an equivalence relation. The reader
is directed to Section 2 and the Appendix for a comprehensive discussion of this
definition.
Let H2 denote the hyperbolic plane. We call a homomorphism
ρ : L Ñ PSL2 pRq
as a projective representation of L, which is also regarded as a circle action:
PSL2 pRq – Isom` pH2 q ãÑ Homeo` pS 1 q.
A projective representation ρ : L Ñ PSL2 pRq is parabolic-free if ρpLq does not
contain nontrivial parabolic elements. A parabolic-free representation is a generalization of a hyperbolic structure without cusps.
Notation 1.1. Let L be a group.

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S. KIM, T. KOBERDA, AND M. MJ

(1) We denote by MpLq the double coset space
MpLq “ OutpLqz HompL, Homeo` pS 1 qq{semi-conjugacy.
(2) We denote by Xproj pLq the set of faithful parabolic-free projective representations of L.
In other words, MpLq is the space of the equivalence classes of circle actions by
L, which in turn can be regarded as the “moduli space” of circle actions by L. The
motivating question for this paper is the following.
Question 1.2. For which finitely generated group L is the image of Xproj pLq in MpLq
nonempty or uncountable? What is the topology of the image of Xproj pLq in MpLq?
We now summarize the principal results of this paper.
1.1. Combination Theorems and Indiscrete Subgroups of PSL2 pRq. Before stating the main combination theorem, we need to introduce some terminology. For a
group L, we let T L denote the set of torsion elements.
A set of representations Λ Ď HompL, PSL2 pRqq is tracially disjoint if every pair
of distinct elements ρ, λ P Λ satisfy
tr2 ˝ρpLzT L q X tr2 ˝λpLzT L q “ ∅.
Recall from our definition that a projective representation that factors as
L

/

PSL„
2 pRq

p

/

PSL2 pRq

is called liftable; here, p is the universal covering map. We will say a subgroup
L ď PSL2 pRq is liftable if so is the embedding L Ñ PSL2 pRq.
Definition 1.3. A finitely generated group L is called flexible if there exists a uncountable, tracially disjoint subset Λ of indiscrete faithful parabolic-free projective
representations. If, furthermore, Λ can be chosen so that each representation in Λ
is liftable, then we say L is liftable-flexible.
Excluding the torsion elements is necessary, since if g P PSL2 pRq has order n in
L then there are only n possible values that tr2 pgq can take. We will see later that
the space MpLq is uncountable for a flexible group L. See Corollary 5.11 for further
implications of flexibility.
We let Innpgq denote the conjugation map
h ÞÑ ghg´1 .
We also use the notation hg “ g´1 hg. Recall a subgroup C ď A is malnormal
if C g X C “ t1u for all g P AzC. Let A be a group and let φ : C Ñ C 1 be an
isomorphism between subgroups of A. We let
A˚φ : CÑC 1 “ A˚φ “ A˚ s

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

5

denote the HNN extension of A amalgamated along the map φ, where s is used as
the stable generator.
Let F and F „ denote the classes of flexible and liftable-flexible groups, respectively. The following result is the principal combination theorem which we
establish in this paper.
Theorem 1.4 (Combination Theorem for Flexible Groups). Let H “ F or F „ .
(1) Infinite cyclic groups are in H .
(2) If H ď L P H and rL : Hs ă 8, then H P H .
(3) If A and B are in H , then so is A ˚ B.
(4) Suppose A, B ď L and C ď A X B satisfy
L “ xA, By P H .
If C is nontrivial malnormal abelian in A and in B, then A ˚C B P H .
(5) Suppose A, C ď L and s P L satisfy
L “ xA, sy P H
and that C and C s are nontrivial malnormal abelian subgroups of A. Assume either
‚ s P ZpCq, or
‚ C and C s are not conjugate in A.
Then A˚φ P H , where φ “ Innpsq : C Ñ C s .
(6) If A is in F and B is a finite cyclic group, then A ˚ B P F .
The groups in F „ are torsion-free, so part (6) does not hold for F „ .
Remark 1.5. If L is a torsion-free flexible group and if C is a maximal abelian
subgroup of L, then C is malnormal. This is because for a hyperbolic maximal
abelian subgroup µ ď PSL2 pRq, the set Npµqzµ consists of elements with order
two.
By a surface group, we will mean the fundamental group of a closed orientable
hyperbolic surface. Although looks technical, the condition (5) is precisely required
when one considers a decomposition of a surface group by a nonseparating simple
closed curve. A relatively straightforward application of Theorem 1.4 above is for
a double of a flexible group. For a group A and a subgroup C, we denote by A ˚C Ā
the amalgamation of two copies of A along IdC . We write A˚C “ A ˚C pC ˆ Zq. By
putting A “ B or s “ 1 in the Combination Theorem, we obtain:
Corollary 1.6. Let A be a group and let C be a malnormal abelian subgroup of A.
If A is flexible or liftable-flexible, then so are A˚C and A ˚C Ā.
With some more work, we obtain the following very general result about circle
actions of limit groups. Recall that a limit group is a finitely generated group which
is fully residually free.

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S. KIM, T. KOBERDA, AND M. MJ

Corollary 1.7. Every nontrivial limit group is liftable-flexible.
A finite type hyperbolic 2–orbifold S , and its fundamental group, are called splittable if some simple closed curve splits S into hyperbolic pieces (Definition 4.8).
For instance, all surface groups and Fuchsian groups satisfying χ ď ´1 are splittable.
Flexibility is important to us because it allows us to construct many dynamically
distinct actions of groups on the circle. Within the context of groups which “classically” act on the circle through discrete projective representations, we have the
following.
Theorem 1.8.
(1) Finite type splittable Fuchsian groups are flexible.
(2) Finite type torsion-free Fuchsian groups are liftable-flexible.
In the course of establishing Theorem 1.4, we prove several other results which
are of independent interest. Probably the most important of these are several general
versions of Baumslag’s Lemma.
Classically, Baumslag’s Lemma [6] is a criterion for certifying that elements of
free groups do not reduce to the identity, and is fundamental in the study of limit
groups. We prove several generalizations of Baumslag’s Lemma, which we will not
state here for the sake of brevity. These include a Topological Baumslag Lemma
for group actions on an arbitrary Hausdorff topological space (Lemma 3.6), Projective Baumslag Lemma which applies to subgroups of PSL2 pCq (Lemma 3.7), and
Discrete Baumslag Lemma which applies to acylindrically hyperbolic groups and
mapping class groups of surfaces (Theorem 3.9 and Corollary 3.13). Baumslag’s
Lemmas coupled with a Baire category argument for representation varieties allows
us to prove a number of combination theorems for faithful (but possibly indiscrete)
representations, including Theorem 1.4.
1.2. Uncountable Families of Exotic Group Actions on the Circle. The combination theorems discussed above are the technical tools we use to construct large
families of actions of “classical” groups on the circle with essentially different dynamical behaviors, which is to say that the actions are not semi–conjugate to each
other.
We distinguish between various semi-conjugacy classes of actions by estimating
their rotation spectra. Recall that if
f P Homeo` pS 1 q
then f has a well-defined rotation number rotp f q, which is given by lifting f to
fr P Homeo` pRq

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

7

and computing the limit
frn p0q
pmod Zq.
nÑ8
n
The rotation spectrum of a circle action of L is given by
lim

rot ˝ρpLq “ trot ˝ρpgq : g P Lu Ď R.
One can also define the marked rotation spectrum of ρ by
rot ˝ρ “ tpg, rot ˝ρpgqq : g P Lu P RL .
The rotation number is a semi-conjugacy invariant (Theorem 2.12). Hence the
rotation spectrum of an action is an equivalence invariant of an action, and the
marked rotation spectrum is a semi-conjugacy invariant of an action. The notion
of rotation spectrum has attracted much attention in the literature; see [22, 44] for
instance, as well as the discussion in Subsection 1.6.5 below.
If ρ is a circle action of L, then its Euler cocycle ρ˚ eu defines a class ρ˚ eub P
2
Hb pL; Zq; see Theorem 2.6. We treat the following consequence of flexibility in
Section 5.
Corollary 1.9. If L is a flexible group, then there exists a uncountable subset Λ of
Xproj pLq such that the following hold.
(i) The natural map Λ Ñ MpLq defined by ρ ÞÑ rρs is an injection.
(ii) The image of Λ is Z–linearly independent in Hb2 pL; Zq.
(iii) If, furthermore, L is liftable-flexible, then Λ maps into the kernel of
Hb2 pL, Zq Ñ H 2 pL, Zq.
In [74], K. Mann shows that if L is a surface group and if Z is a component of
the representation space
X “ HompL, Homeo` pS 1 qq
which contains a discrete faithful representation L Ñ PSL2 pRq, then Z consists
of one semi-conjugacy class. We exhibit an opposite, “local flexibility” phenomenon for indiscrete representations of Fuchsian groups (possibly with torsion). See
Section 4.3 for the definition of an almost faithful path.
Corollary 1.10 (See Theorem 4.11). Let L be a Fuchsian group and let ρ0 P
Xproj pLq be indiscrete. Then for each countable set W Ď R, there exists an almost
faithful path tρt utPr0,1s in HompL, PSL2 pRqq such that
tr2 ˝ρ1 pLzT L q X W “ ∅.
In other words, every open neighborhood of ρ0 contains uncountably many, pairwise non-semi-conjugate representations from Xproj pLq. In particular, the point
rρ0 s P MpLq is an accumulation point of the image of Xproj pLq and this gives a

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S. KIM, T. KOBERDA, AND M. MJ

partial answer to the second half of Question 1.2. Note that one cannot further
require ρ s semi ρt for all 0 ď s ă t ď 1; see Proposition 4.12.
It is an open question whether or not the space X above has infinitely many
connected components [74]. The representations in
Y “ HompL, PSL2 pRqq
with Euler number 0 form a connected component [48] of Y, and we show that this
component contains uncountably many equivalence classes, as follows easily from
Corollary 1.9. More generally, we have the following.
Corollary 1.11. Let L be a finite type splittable Fuchsian group. Then some path–
component of
HompL, Homeo` pS 1 qq
contains uncountably many distinct equivalence classes of faithful actions.
Corollary 1.11 is complemented by work of M. Wolff. See Subsection 1.6.3
below.
We note another consequence of Corollary 1.9. Suppose f : L Ñ Z is a map. We
define the coboundary of f as
B f pg, hq “ f pgq ` f phq ´ f pghq
and the defect of f as
Dp f q “ }B f }8 “ supt|B f pg, hq| : g, h P Lu.
We say f is a quasi-morphism if Dp f q ă 8. A minimal quasi-morphism is an
integer–valued subadditive defect–one quasimorphism. See Section 5 for details.
Corollary 1.12. If L is liftable-flexible, then there exists a set Λ of minimal quasimorphisms such that
trρs P HQMpL; Zq : ρ P Λu
is uncountable and linearly independent.
Here, HQMpL; Zq denote the set of quasi-morphisms from L to Z, quotiented
by homomorphisms and bounded maps. See Section 2 for a more detailed discussion of quasimorphisms, circle actions, and bounded cohomology. We note here a
corollary for limit groups.
Corollary 1.13. Every limit group admits a uncountable, Z–linearly independent
set of minimal quasi-morphisms.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

9

1.3. An Axiomatic Approach to Combination Theorems. In Section 6 below,
we develop an axiomatic framework for combination theorems for group representations where the target group is a general Baire topological group G. A far more
concrete setup is discussed in Section 5, and the axioms in Section 6 might seem
technical and (overly) abstract to the reader at the first reading. For this reason, we
have chosen the present expository mode of providing many examples in Section 5
and motivating the abstract axiomatic nature of Section 6 that follows.
The notion of a trace is replaced by a more general notion of a tracial structure. The most important concepts encoded in a tracial structure (Definition 6.1)
are a generalized trace (a continuous function on conjugacy classes) and a notion
of genericity, in the form of Bausmlag Lemma. The generalized trace is simply a
continuous map
ϕ: G Ñ S ,
where S is a topological space, and where ϕ is constant on conjugacy classes. The
notion of generalized parabolic elements in this situation is defined accordingly:
P “ ϕ´1 ˝ ϕp1q,
i.e. the generalized parabolic elements are precisely those whose traces agree with
that of the identity with respect to the tracial structure given by ϕ. Using this structure, we will prove an axiomatic combination theorem (Theorem 6.6), which generalizes Theorem 1.4.
1.4. Flexibility and Rigidity. Among group actions on the circle with a given
rotation spectrum, there appears to be a mixture of rigidity and flexibility, as we
illustrate with the following results.
On the one hand, projective representations impose a high degree of rigidity.
The following fact is relatively well–known (see for instance Katok’s book [61],
Theorem 2.5.4):
Proposition 1.14 (cf. Theorem 8.3). Let L be a finitely generated group and let φ
be a faithful projective action of L with rot ˝φpLq “ t0u. Then φpLq is conjugate to
a discrete subgroup of PSL2 pRq. In particular, either:
(1) The group L is the fundamental group of a closed orientable hyperbolic surface S , and φ corresponds to a complete hyperbolic structure on S ; in particular, up to an automorphism of L, there are only two conjugacy classes
of such actions;
(2) The group L is free, and there are finitely many equivalence classes of such
projective L–actions on S 1 , two for each homeomorphism type of a surface
S with π1 pS q – L.
On the other hand, even marked rotation spectra do not form a complete equivalence invariant of surface group actions:

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S. KIM, T. KOBERDA, AND M. MJ

Theorem 1.15 (cf. Theorem 8.25). There exists a faithful action φ of a finitely
generated nonabelian free group L on S 1 with (marked) rotation spectrum t0u and
pkq
such that φ is not semi-conjugate to a subgroup of PSL2 pRq, for any k ě 1.
pkq

Here, PSL2 pRq denotes the connected k–fold cover of PSL2 pRq. In particular,
the action of the free group in Theorem 1.15 is not semi-conjugate to a projective
action.
The proof of Theorem 1.15 is remarkable in that it uses several nontrivial facts
about hyperbolic 3–manifolds which fiber over the circle. We remark furthermore
that Theorem 1.15 considers flexibility of actions within a class of actions with a
fixed rotation spectrum. Thus, this result is one of the several in this paper which
are completely different from (as opposed to complementary to) the other results
in literature which study flexibility and rigidity of actions via rotation numbers
(cf. [22, 26, 44, 73], for instance).
K. Mann has pointed out a statement analogous to Theorem 1.15 for surface
groups, which appeared in her thesis [71]. Her argument shares certain features
with the proof of Theorem 1.15 given in this paper, though the crux of her argument
(which relies on Thurston norm and maximal Euler classes) and motivations for
considering the problem in the first place are different from those of the authors.
Result similar to Theorem 1.15 have also been produced by Barbot–Fenley [4],
using pseudo-Anosov flows. See Subsection 1.6.8 below.
Hyperbolic 3–manifold groups contain a profusion of closed surface groups,
both quasi–Fuchsian and geometrically infinite, and fibered hyperbolic 3–manifold
groups admit natural circle actions, i.e. the universal circle action (see Section 7
and Subsection 8.3. We have the following semi–conjugacy characterization of
such surface group actions:
Theorem 1.16 (cf. Theorem 8.17). Let G be a quasi–Fuchsian subgroup of a closed
fibered hyperbolic 3–manifold group. Then the universal circle action of G is semi–
conjugate to a Fuchsian group action.
Theorem 1.17 (cf. Theorem 8.23). Let M be a fibered hyperbolic 3–manifold with
a fixed fibration with fiber S , and let G be a fiber subgroup of a fibration of M which
lies in the same fibered face of the Thurston norm ball as S , up to multiplication by
´1. Then the universal circle action of G is conjugate to a Fuchsian group action.
As mentioned above, K. Mann proved a complement to Theorem 8.23 in her
thesis [71], proving that if G lies in a different fibered face of the Thurston norm
ball then G is not semi–conjugate to a Fuchsian group action.
The following result exhibits exotic actions of nonabelian free groups on the
circle, among other exotic group actions:

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

11

Theorem 1.18 (cf. Theorem 8.26). Let F be a nonabelian free group. Then there
exists a faithful C 8 action of F on S 1 which is not semi-conjugate to the action of
any transitive, finite–dimensional, connected Lie subgroup of Homeo` pS 1 q.
1.5. Mapping Class Groups. We conclude our discussion with a study of distinct
semi-conjugacy classes of mapping class group actions on the circle (see Section 7).
Recall that if S is an orientable surface, its mapping class group ModpS q is the
group orientation–preserving homeomorphisms of S up to isotopy. We will always
assume that S has negative (though finite) Euler characteristic. A mapping class
group is called bounded if BS ‰ ∅.
Proposition 1.19 (cf. Proposition 7.2). A bounded mapping class group has at least
two inequivalent actions on the circle.
1.6. Notes and References.
1.6.1. Circle Actions and Quasi-morphisms. Among the original motivations for
this paper was a question posed to the authors by M. Bestvina and K. Fujiwara,
namely whether one can explicitly produce distinct, subadditive, integer–valued,
defect–one quasimorphisms of a free group or a surface group. This paper answers
their question by producing uncountable families of examples which come from
quasimorphisms of the corresponding groups.
It was pointed out to the authors by D. Calegari that defect–one (possibly not
subadditive) quasimorphisms from a given quasimorphism is relatively straightforward, so that producing uncountably many linearly independent ones is not such a
huge generalization. In this paper however, the quasi-morphisms we produce are
subadditive and defect–one, these two properties being not so trivial to simultaneously guarantee, since such a quasimorphism (as a second bounded cohomology
class) corresponds to a semi-conjugacy class of a circle action that lifts to the real
line.
1.6.2. Generalizations to Other Semi-simple Algebraic Groups. Many of the results about PSL2 pRq which we discuss in this paper generalize suitably to other
semi-simple algebraic groups. When discussing algebraic groups, we concentrate
on PSL2 pRq since we are primarily interested in the dynamical picture. Indeed,
lattices in non-split higher rank semi-simple linear algebraic groups admits no interesting actions on the circle [20, 103, 46]. Thus, while the continuous version of
Baumslag’s Lemma (see Lemma 3.7 below) admits a generalization to other algebraic groups, for instance, we avoid making such generalizations for the sake of
brevity and clarity, especially since the dynamical consequences would be rather
limited for us. We reiterate, however, that this paper develops a representation–
theoretic perspective for studying dynamics in low dimensions which fits into a
larger context of indiscrete subgroups of semi-simple algebraic groups.

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S. KIM, T. KOBERDA, AND M. MJ

1.6.3. Towards a Teichmüller Theory for Indiscrete Representations. Most classical constructions of free group actions on various spaces, and more generally of
free product actions, rely on some version of the ping–pong lemma [33, 100, 67].
The novelty of the approach given here is in the construction of dense subgroups of
PSL2 pRq with various exotic properties, which are situations to which ping–pong is
unadaptable. We note that, even though we are concentrating on dense subgroups
of PSL2 pRq, our constructions are generally informed by and often rely on classical
discrete subgroup constructions.
The rotation spectrum of a representation is an equivalence invariant of the representation, which plays a role similar to the length spectrum for a discrete representation of a group L into PSL2 pRq. Classical Teichmüller Theory gives conditions
under which length spectra determine a surface group representation up to conjugacy (see [91], for instance). In the situation of indiscrete actions, the length
spectrum for hyperbolic elements in the image will be dense inside of R` . Furthermore, it seems likely that rotation spectrum alone will not generally determine the
semi-conjugacy class of a faithful action:
Question 1.20. Let L be a surface group and let ρ0 , ρ1 be projective representations
of L on S 1 . If rot ˝ρ0 pLq “ rot ˝ρ1 pLq, under what conditions is ρ0 is equivalent to
ρ1 ?
We make two further remarks about rigidity of actions. First, there is a sense in
which the marked rotation spectrum (with some extra data) does in fact determine a
circle action up to semi-conjugacy, which is given by Matsumoto’s Theorem [77].
We will give an exact statement of this result in Theorem 2.12.
Secondly, the work of M. Wolff gives an answer to Question 1.20 in the case of
marked rotation spectra:
Theorem 1.21 (M. Wolff [104], see also [105]). Let ρ0 and ρ1 be projective representations of a finitely generated group L. If ρ0 is non-elementary and indiscrete,
and if for all g P L we have
rot ˝ρ0 pgq “ rot ˝ρ1 pgq,
then ρ0 and ρ1 are conjugate in PSL2 pRq.
In the case of unmarked rotation spectra, it seems that relatively few properties of
a projective group action are determined by the unmarked rotation spectrum [75].
1.6.4. Dense Limit Subgroups of Algebraic Groups. The work in Section 4 is closely
related to, and partially inspired by, the work of Breuillard–Gelander–Souto–Storm [16]
and of Barlev–Gelander [5] in which those authors studied the relationship between
dense free subgroups and dense limit subgroups of semi-simple algebraic groups.
In the case of PSL2 pRq actions, we show not only the existence of dense non-free

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

13

limit subgroups, (thus recovering the main results of [16] and [5] in this case) but in
fact an enormous abundance of such subgroups, as witnessed by our constructions
of uncountable, tracially disjoint (hence inequivalent) families of representations in
PSL2 pRq.
Perhaps the most significant conceptual difference between the present work and
that of [16] and of [5] is the form in which Baumslag lemmas are stated, proven
and used. In [16] and [5], the fundamental combinatorial group theory tool is the
Baumslag lemma for free groups. In the present work, we use Projective Baumslag
Lemma adapted to the ambient Lie group, and Topological and Discrete Baumslag
Lemma adapted to more general setups in which north–south dynamics is present.
It is thus that we are able to apply our machinery to a larger array of groups in order
to deduce the dynamical consequences outlined above.

1.6.5. Relationship to the Work of Calegari and Calegari–Walker. Various rigidity and flexibility phenomena concerning rotation numbers for groups acting on
the circle have been studied by many authors as we have indicated above, and will
indicate further below. As for the ideas of this paper which concern rotation numbers, the most closely related work is probably the work of Calegari in [22] and
Calegari–Walker in [26].
In [26], Calegari and Walker study actions of the rank two free group F2 on the
circle, and the various rigidity and rationality phenomena that can be described
therein. For instance, if two free generators act with rational rotation numbers r
and s then the supremum Rpw, r, sq of the possible rotation numbers achieved by
each fixed positive word w P F2 in those generators is again rational. Moreover, the
denominator of Rpw, r, sq is well–controlled and computable. The value Rpw, r, sq
is locally constant from the right for a positive w and r, s P Q.
In [22], Calegari considers (among other things) the following problem: let θ P
R{Z. Is there a finitely presented group L acting on S 1 and an element α P L such
that Yρ rot ρpαq “ t˘θ, 0u where ρ varies over all possible actions of L on the
circle? If the answer is yes then α is called a forceable number. Calegari shows
that the set of forceable numbers is very large: lifting the set of forceable numbers
to R, one obtains an infinite dimensional rational vector subspace of R. Moreover,
the set of forceable numbers contains infinitely many algebraically independent
transcendental numbers.
Perhaps the most significant difference between this work and that of Calegari [22]
is the fact that in the present work we study possible rotation numbers within the
context of actions of a single group, i.e. we do not control all actions of a given
finitely presented group on the circle. Our purpose is the opposite, in fact: we use
large diversity in rotation numbers to deduce the existence of many actions.

14

S. KIM, T. KOBERDA, AND M. MJ

Moreover, as noted above, we expend a considerable effort to produce inequivalent actions of groups on the circle which are not distinguished by rotation numbers,
which is to say flexibility of actions with a fixed rotation spectrum.

1.6.6. Dense Sets of Faithful Projective Surface Group Actions. In [34], Deblois
and Kent show that for a surface group L, the set of faithful representations inside
the representation variety
HompL, PSL2 pKqq
is dense in the classical analytic topology, where K P tR, Cu. In fact, they show
that the set of faithful representations is a dense Gδ in the classical topology. In
their proof, they even use a fact which may be regarded as a transcendental/number
theoretic Baumslag Lemma in order to establish the properness of vanishing loci in
the representation variety, though the authors do not call it by that name. The work
of Deblois–Kent in the case of K “ R can be used to construct many non-conjugate
actions on the circle (see [72]), though questions of inequivalences of these actions
are not addressed.
The purpose of Deblois and Kent was to prove a conjecture of W. Goldman,
namely that the set of faithful representations is dense in the representation variety. As such, dynamical considerations did not enter into their discussion. On the
one hand, the present work builds a more general framework for studying exotic
group actions on the circle. As a consequence of developing such a framework,
we deduce the existence of an abundance of exotic circle actions, not just for surface groups but also for free groups, limit groups, and many other groups beyond
the scope of the methods used in [34]. On the other hand, this paper does not recover the Deblois–Kent result, since their considerations apply to all components
of the representation variety, whereas the topology of the representation variety is
of secondary importance to us.
Deblois–Kent’s methods are geometric in the sense that they rely essentially on
the geometrization of surfaces. Our approach is a combination of a representation–
theoretic and an algebro–geometric approach which is informed by ideas from geometry. Thus, we are able to deduce dynamical consequences for groups beyond
those appearing in two–dimensional hyperbolic geometry. The particular features
of our work which do not fall under the purview of [34] include but are not limited
to combination theorems for indiscrete subgroups of PSL2 pRq, general Baumslag
Lemmas, and applicability to limit groups. Moreover, we construct a path in the
representation space of a Fuchsian group such that almost every point on the path is
a faithful representation (cf. Theorem 4.11), which does not follow formally from

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

15

Deblois–Kent. Finally, while the present article focuses on projective representations of groups, a large part of the paper concerns general smooth and even topological actions of groups on the circle. The following question attempts a natural
generalization Deblois–Kent result in our setting:
Question 1.22. Let L be a nonabelian limit group. Is the set Xproj pLq very general
in HompL, PSL2 pRqq?
In Question 1.22, the terminology very general is a certain Baire category condition. See Section 4.1 for a precise discussion.
1.6.7. Projective Actions Versus Analytic Actions. Questions similar to those addressed in this paper are studied in [1] in the context of analytic group actions on
the circle. A group action on the circle is called analytic if every element acts by
a homeomorphism which is locally given by a convergent power series in the local parameter. The group PSL2 pRq acts on the circle by analytic diffeomorphisms,
though not every analytic action of a group corresponds to a subgroup of PSL2 pRq.
In [1], the authors of that paper define a combinatorial invariant called a topological skeleton, which plays the role of a Markov partition, and which allows them to
classify virtually free group actions on the circle which are analytic.
Among the consequences of the main results of [1] are certain rigidity results for
semi-conjugacy of actions with a exceptional minimal set. They also obtain results
which are related to Theorem 1.15. Namely, they produce certain locally discrete
finitely generated free groups of analytic diffeomorphisms of S 1 which have exotic dynamics. For example, they produce a two–generated subgroup of analytic
diffeomorphisms which acts minimally, which is free of rank two, where one generator acts like a hyperbolic element of PSL2 pRq, and where one generator has two
parabolic fixed points. In particular, such an action cannot be semi-conjugate to a
projective action. The authors show that the rotation spectrum of the actions they
consider are finite.
Thus, the authors of [1] produce examples of a similar ilk to those furnished in
Theorem 1.15, in the case of free groups. Their methods are very different from
those employed in this paper, and they do not extend to analytic surface group
actions.
1.6.8. Non–Fuchsian Exotic Actions. Groups which act on the circle in ways that
resemble Fuchsian group actions but which are not projective actions (as in Theorem 1.15) have been studied by several authors, as has been pointed out by M.
Triestino. The ideas find their origin in Thurston [97], which were then clarified
and further developed by Brooks in his appendix to Bott’s article [12], and then
further studied by Tsuboi [101].

16

S. KIM, T. KOBERDA, AND M. MJ

The work in [1] suggests that locally discrete analytic virtually free group actions should be semi-conjugate into a certain overgroup studied by Tsuboi in [101].
As discussed above, other examples of non-Fuchsian exotic actions appear in [71]
and [4].
1.6.9. Groups Without Exotic Actions. There are many groups which naturally act
on the circle with various degrees of regularity, but which do not admit any (or at
least very few) “exotic” actions. Precisely, there are finitely generated groups
G ď Diff k pS 1 q ď Homeo` pS 1 q
such that if k " 0 (generally k ě 2 suffices), then a faithful C k action of G on
Homeo` pS 1 q has periodic points and is therefore semi-conjugate to an action of
some finite cyclic group. Natural classes of groups with such properties lie in the
class of right-angled Artin groups. For an explicit example, one can consider G “
F2 ˆ F2 . Standard applications of Denjoy–Koppell type arguments show that any
faithful C 2 action of G has a periodic point (cf. [63]). We remark that in general,
right-angled Artin groups admit no faithful actions on the circle of C 2 or higher
regularity (see [3, 63], also [39, 58]).
It seems that the degree of regularity one requires of group actions can have a
significant effect on the existence or nonexistence of exotic actions. For instance,
note that every right-angled Artin group occurs as a subgroup of the mapping class
group of some surface with boundary (see [68] and the references therein), and
these mapping class groups do admit exotic actions (see Section 7 below), but no
faithful mapping class group action is conjugate to a C 2 action by [3].
1.6.10. Mapping Class Groups. The fact that the mapping class group of a surface
with a marked point embeds into Homeo` pS 1 q is an old result of Nielsen (see [87,
88, 89], also [28, 53]). Such actions for mapping class groups with boundary on the
circle and on the real line were studied by Handel–Thurston [53]. The existence of
exotic actions of mapping class groups (i.e. ones not semi–conjugate to Nielsen’s
original action) was posed as a question by Farb [38], and has been also studied by
Bowditch–Sakuma [14].
1.7. Outline of the Paper. We have striven to make the present article as self–
contained as possible. In Section 2 we gather various cohomological facts about
circle actions, where an object of central importance is the bounded Euler class
of an action. We single out Theorem 2.12 for the attention of the reader, since it is
difficult to find a definitive statement of the equivalence of all the various definitions
of semi-conjugacy in the literature.
Section 3 contains the various incarnations of Baumslag’s Lemma, which is one
of the main technical tools of the paper.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

17

Section 4 produces uncountable families of inequivalent Fuchsian and nontrivial
limit group actions on the circle, all arising from indiscrete subgroups of PSL2 pRq.
This discussion generalizes in the form of a combination theorem in Section 5.
Implications of flexibility and liftability for limit groups and quasi-morphisms are
discussed here.
Section 6 develops a more axiomatic setup for the discussion in Section 4 and
in Section 5, and establishes indiscrete combination theorems again for all limit
groups and in the setup where the target group is a general Baire topological group.
The logical dependence of the sections is as follows. Section 2 is basic but can be
skipped by readers familiar with circle actions and bounded cohomology. Section A
complements Section 2 by giving details and proofs. Section 5 relies on Sections 3
and 4. Section 6 is logically independent from the other sections, but is conceptually
informed by Sections 4 and 5.
2. Preliminary
2.1. Actions on the Circle. Ghys employed cohomological methods to “study the
dynamics” [45] of discrete group actions on the circle. More precisely, he defined a
relation between group actions he called semi-conjugacy and exhibited a correspondence between cohomology classes and semi-conjugacy classes; see Theorem 2.6.
Recently, several other closely related notions of semi-conjugacy were suggested
in the literature [24, 19, 73]. These modifications were aimed at converting semiconjugacy into an equivalence relation and further, to have a powerful characterization of circle actions via cohomology classes. For a complete reference on bounded
cohomology of discrete groups, the reader is directed to Frigerio’s recent book [42],
which includes a discussion of second bounded cohomology and circle actions.
A reconciliation of these notions of semi-conjugacy is summarized in Theorem 2.12 below. We refer the reader to the Appendix and the cited references therein
for proofs of equivalence. A different and more detailed exposition of many (but
not all) of these equivalent notions has been given in [19]. The ideas go back to
Ghys’ original article [45], and to [24, 19, 73]. The reader is also directed to [21].
We remark that throughout this section, we make no claims as to originality.
Throughout this section, we let L be a countable group. For r P R, we let T prq
denote the translation by r on R or on S 1 . For brevity, we denote
T “ T p1q : R Ñ R.
Rotation number and Euler class
Let us recall the following well-known terminology. For f P HomeoZ pRq, we define
the translation number
f n pxq ´ x
rot„ p f q “ lim
,
nÑ8
n

18

S. KIM, T. KOBERDA, AND M. MJ

which is independent of the choice of x P R. The Poincaré’s rotation number of
f P Homeo` pS 1 q is defined as
rotp f q “ rot„ p f˜q pmod Zq
where f˜ P HomeoZ pRq is an arbitrary lift of f . Then rot : Homeo` pS 1 q Ñ S 1 is
a continuous homogeneous class function. That is, the map rot is continuous with
respect to the usual (uniform) topology on Homeo` pS 1 q, and
rotp f n q “ n rotp f q,

rotpg f g´1 q “ rotp f q

for f, g P Homeo` pS 1 q and n P Z. The readers may refer to [86] for more details.
For each x P R, there exists a unique section
s x : Homeo` pS 1 q Ñ HomeoZ pRq
satisfying the following for all f P Homeo` pS 1 q:
s x p f qpxq P rx, x ` 1q.
We can define the Euler cocycle based at x, denoted as
eu x : Homeo` pS 1 q ˆ Homeo` pS 1 q Ñ t0, 1u,
by the following condition:
s x p f qs x pgq “ T peu x p f, gqqs x p f gq.
We often write s “ s0 and eu “ eu0 .
It is a routine exercise to see that eu x is indeed a cocycle and takes values in
t0, 1u. Following [19], we denote the corresponding cohomology classes
eu “ reu x s P H 2 pHomeo` pS 1 q; Zq,

eub “ reu x s P Hb2 pHomeo` pS 1 q; Zq.

The classes eu and eub are called the Euler class and the bounded Euler class,
respectively. These classes are independent of the choice of x P R. Moreover,
eu P H 2 pHomeo` pS 1 q; Zq
corresponds to the universal central extension
1

/

xT y

/

HomeoZ pRq

p

/

Homeo` pS 1 q

/

1

For each ρ P HompL, Homeo` pS 1 qq, we have the pull-back of the Euler cocycle
ρ˚ eupa, bq “ eupρpaq, ρpbqq,
which in turn yields cohomology classes
ρ˚ eu P H 2 pL; Zq,

ρ˚ eub P Hb2 pL; Zq.

The rotation number is determined by the Euler cocycle:

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

19

Lemma 2.1. For each g P Homeo` pS 1 q, we have
n
1ÿ
rotpgq “ lim
eupg, gk q mod Z.
nÑ8 n
k“1
The proof is given in Lemma A.2.
Semi-conjugacy
Recall a pull-back diagram of groups
/

L


A
i



j

/C
B
or the group L in the above diagram, is called a fiber product. Given maps i : A Ñ C
and j : B Ñ C, a fiber product exists and unique up to isomorphism. Concretely,
the following subgroup of A ˆ B is a fiber product:

L “ tpa, bq P A ˆ B : ipaq “ jpbqu.
Let X and Y be topological spaces, and let
ρ0 P HompL, HomeopXqq,

ρ1 P HompL, HomeopYqq.

We will write ρ0 ěh ρ1 if there exists a (possibly discontinuous) map h : X Ñ Y
such that the following diagram commutes for all g P L:
X


ρ0 pgq

h
ρ1 pgq

/

X


h

/Y
Y
We say h semi-conjugates ρ0 to ρ1 .
The following definition is due to Ghys [46], with a slight modification by Mann [73].

Definition 2.2.

(1) We say two actions
ρ0 , ρ1 P HompL, Homeo` pRqq

are semi-conjugate if there exists a proper nondecreasing map h : R Ñ R
such that ρ0 ěh ρ1 .
(2) We say two actions
ρ0 , ρ1 P HompL, Homeo` pS 1 qq
are semi-conjugate, and write ρ0 „semi ρ1 , if there exist
‚ a group L̃ and a homomorphism q : L̃ Ñ L,
‚ two semi-conjugate actions in the sense of part (1):
ρ̃0 , ρ̃1 : L̃ Ñ HomeoZ pRq,

20

S. KIM, T. KOBERDA, AND M. MJ

‚ a commutative diagram
7

1

/

Z “ xT y

/

q

L̃


/

ρ̃i

HomeoZ pRq

p

/

L


ρi

Homeo` pS 1 q

%/

1

for each i “ 0, 1 such that the bottom row is the natural universal central
extension and such that the middle square is a fiber product.
Lemma 2.3. The semi-conjugacy is an equivalence relation on HompL, Homeo` pRqq
and also on HompL, Homeo` pS 1 qq.
Proof. The statement is an immediate consequence of Lemma A.1.



Definition 2.4. Let M “ R or M “ S 1 , and let h : M Ñ M be a (possibly discontinuous) map.
(1) For M “ R, we say h is a monotone degree one map if h is nondecreasing
and if hpx ` 1q “ hpxq ` 1 for each x P R.
(2) For M “ S 1 , we say h is a monotone degree one map if h has a lift to R
which is monotone degree one.
Remark 2.5.

(1) A nondecreasing map h : R Ñ R is proper if and only if
lim hpxq “ ˘8.

xÑ˘8

In particular, we require h to be non-constant.
(2) In Definition 2.2 (2), a proper nondecreasing map h satisfying ρ̃0 ěh ρ̃1
must be monotone degree–one. For, we should have
h ˝ ρ̃0 pT q “ ρ̃1 pT q ˝ h.
(3) Some authors use the terminology that ρ0 is “semi-conjugate” to ρ1 if there
exists a continuous monotone degree one map h : M Ñ M such that ρ0 ěh
ρ1 , where M “ R or M “ S 1 . This notion of “semi-conjugacy” is not an
equivalence relation.
2
We let Ht0,1u
pL; Zq be the set of cohomology classes in Hb2 pL; Zq that can be
represented by t0, 1u–valued cocycles. In particular, we have
2
eub P Ht0,1u
pHomeo` pS 1 q; Zq.

An algebraic characterization of a semi-conjugacy class is given as follows.
Theorem 2.6 ([46, 47]). There exists a one-to-one correspondence:
2
HompL, Homeo` pS 1 qq{ „semi Ñ Ht0,1u
pL; Zq,

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

21

given by
rρs ÞÑ ρ˚ eub .
Limit set of a circle action
A topological action of a group L on a space X is called minimal if every L–orbit
is dense in X. If L ď Homeo` pS 1 q, then L acts minimally on its limit set, which is
defined as
č
ΛpLq : “
L.x.
xPS 1

There is a trichotomy for ΛL as below; see [86, Theorem 2.1.1].
(i) ΛpLq “ S 1 ; in this case, L is minimal.
(ii) ΛpLq is a Cantor set, which is called an exceptional minimal set.
(iii) ΛpLq is finite (possibly empty); in this case, there exists a finite L–orbit.
Remark 2.7. When we have (i) or (ii), the set ΛL is the smallest nonempty minimal
L–invariant closed subset of S 1 , and coincides with the notion of the minimal set in
the literature. In the case (iii), a minimal set means a finite orbit.
For a representation ρ : L Ñ Homeo` pS 1 q, we will also use the notation
Λpρq “ Λ ˝ ρpLq “ ΛpρpLqq.
Lemma 2.8. Let H ď G ď Homeo` pS 1 q be groups such that ΛpHq ‰ ∅. Assume
either
(a) H E G, or
(b) G is finitely generated and rG : Hs ă 8.
Then ΛpHq “ ΛpGq.
Proof. Note that G has no finite orbit, and ΛpHq Ď ΛpGq. Suppose (a). Then for
each g P G we have
č
č
gΛpHq “
gH.x “
gHg´1 .gx “ ΛpHq.
xPS 1

xPS 1

By the minimality of ΛpGq, we see ΛpGq Ď ΛpHq.
We now assume (b). There exists a finite index normal subgroup H0 of G such
that H0 ď H. By part (a), we have
ΛpGq “ ΛpH0 q Ď ΛpHq.



Blow-up and minimalization
In Definition 2.2, it is not required to have a continuous semi-conjugating map
h : R Ñ R such that ρ̃0 ěh ρ̃1 . It is often desirable to have notions of semiconjugacy defined by surjective (hence, continuous) semi-conjugating maps. These
notions can be defined by blowing-up and minimalization.

22

S. KIM, T. KOBERDA, AND M. MJ

Definition 2.9. Let ρ and ρ̄ be circle actions of L.
(1) We say ρ is a blow-up of ρ̄ if ρ ěh ρ̄ for some surjective monotone degree
one map h on S 1 .
(2) We say ρ̄ is a minimalization of ρ if one of the following holds:
(i) ρpLq has a finite orbit, and ρ̄ “ T ˝ rot ˝ρ;
(ii) ρ is a blow-up of ρ̄, and ρ̄ is minimal.
Remark 2.10. The cases (i) and (ii) are mutually exclusive.
A minimalization can be regarded as a “standard form” of a circle action. The
proof of the following lemma is given in the Appendix, after Lemma A.6.
Lemma 2.11. Every circle action admits a minimalization, which is unique up to
conjugacy.
The most interesting case of the above lemma is when the limit set is a Cantor
set. In this case, the semi-conjugating map h is the Cantor function and also called
as the devil’s staircase function.
For f, g P Homeo` pS 1 q and their arbitrary lifts f˜, g̃ P HomeoZ pRq, Matsumoto [77]
defined the canonical Euler cocycle as
τp f, gq “ rot„ p f˜g̃q ´ rot„ p f˜q ´ rot„ pg̃q.
Then τ is a continuous real-valued 2-cocycle and independent of the choice of lifts.
Using our notation for a section
s “ s0 : Homeo` pS 1 q Ñ HomeoZ pRq,
we can more succinctly write
τ “ ´s˚ pB rot„ q.
The following Theorem reconciles some of the notions of “semi-conjugacy” in
the literature.
Theorem 2.12. Let L be a countable group. For two circle actions ρ0 , ρ1 of L, the
following statements are all equivalent.
(1) ρ0 and ρ1 are semi-conjugate.
(2) ρ0 and ρ1 have a common blow-up.
(3) ρ0 and ρ1 have a common minimalization.
(4) ρ0 and ρ1 have the same bounded Euler class.
(5) We have rot ˝ρ0 “ rot ˝ρ1 and ρ˚0 τ “ ρ˚1 τ as maps on L and on L ˆ L,
respectively.
(6) There exists a continuous path
tρt : t P r0, 1su Ď HompL, Homeo` pS 1 qq
such that for each g P L, the value rot ˝ρt pgq is constant on t P r0, 1s.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

23

(7) ρ0 ěu ρ1 and ρ1 ěv ρ0 for some monotone degree one maps u and v on S 1 .
Theorem 2.12 now provides various equivalent definitions of semi-conjugacy. In
the Appendix, we will give a self-contained proof for
p7q ks

+3

p2q
+3

p5q ks

p1q
z

p6q

+3

p3q


p4q

which are the most relevant parts of the theorem for this paper. We will then indicate
references for elementary proofs of (5)ñ(1) and (7)ñ(4), thus establishing all the
equivalence. See also the references in the following remarks. We emphasize again
that in (1) and (7), semi-conjugating maps are not required to be continuous.
Remark 2.13. Mann [73] suggested the condition (1), in order to modify Ghys’ original definition of semi-conjugacy to become an equivalence relation. She also used
the condition (6) while studying connected components of surface groups representations [74, Corollary 7.5 and Proposition 7.6]. The condition (3) naturally stems
from an exposition of Ghys [47]. Matsumoto [77] gave an essentially equivalent
definition to (5). Extensive study on the properties of τ and the rotation numbers
of longer words has been done in the literature; see [57, 85, 26]. The condition
(7) is probably the simplest definition among above, and due to Takamura [96] and
Bucher–Frigerio–Hartnick [19].
Remark 2.14. One can define a weak monotone equivalence as the equivalence relation generated by a ě–relation where the semi-conjugating monotone degree one
map is required to be surjective. This equivalence was originally called “monotone
equivalence” in [25]. The agreement of these two notions of monotone equivalence
was noted in [24, Lemma 2.22], in which book the equivalence of (2) and (4) is also
shown. The agreement between weak monotone equivalence and the condition (7)
can be found in [19].
Corollary 2.15. Let L be a countable group and let ρ0 , ρ1 be two semi-conjugate
circle actions of L. If ρ0 has a finite orbit of cardinality N ą 0, then so does ρ1 .
For a proof, see Lemma A.5 in the Appendix. The equivalence of (1) and (6) in
Theorem 2.12 implies the following.
Corollary 2.16 ([74, Proposition 7.6]). In the space of the circle actions of a finitely
generated group, each semi-conjugacy class is contained in some path component.
We lastly record a simple corollary for Z “ x1y.

24

S. KIM, T. KOBERDA, AND M. MJ

Corollary 2.17. For circle actions ρ0 , ρ1 of xsy – Z, we have ρ0 „semi ρ1 if and
only if
rot ˝ρ0 psq “ rot ˝ρ1 psq.
Minimal quasi-morphisms
For a group G, let QMpL; Zq denote the abelian group of integer–valued quasimorphisms on G. Define
`
˘
HQMpL; Zq “ QMpL; Zq{ BpL; Zq ‘ H 1 pL; Zq
where BpL; Zq consists of bounded maps G Ñ Z.
The defect Dpαq of α P QMpL; Zq is defined by
Dpαq “ sup |Bα| “ supt|αpgq ` αphq ´ αpghq| : g, h P Lu.
If α P QMpL; Zq has defect–one, then α has the smallest possible defect among
nontrivial integral quasimorphisms.
Definition 2.18. Let L be a group. A map α : L Ñ Z is called a minimal quasimorphism if αpgq ` αphq ´ αpghq P t0, 1u for all g, h P L.
In other words, a minimal quasimorphism on L is an subadditive, integer–valued
defect–one quasimorphism. For example, the quasimorphism n ÞÑ rπns is minimal.
We say a circle action ρ of L is liftable if ρ factors as
L Ñ HomeoZ pRq Ñ Homeo` pS 1 q.
This is equivalent to the condition
ρ˚ eub P kerpHb2 pL; Zq Ñ H 2 pL; Zqq.
So if ρ is liftable, then we have an equality of cocycles
ρ˚ eu “ Bα
for some map α : L Ñ Z. Since ρ˚ eu is t0, 1u-valued, we see that
Bαpx, yq “ αpxq ` αpyq ´ αpxyq P t0, 1u.
In other words, α is a minimal quasimorphism. Note
rαs P HQMpL; Zq
maps to ρ˚ eub by the well-known exact sequence [24]:
0 Ñ HQMpL; Zq Ñ Hb2 pL; Zq Ñ H 2 pL; Zq.
By Theorem 2.6, we have the following.
Lemma 2.19. Suppose we have liftable circle actions ρ0 , ρ1 of L. If ρ0 and ρ1 are
not semi-conjugate, then the minimal quasimorphisms corresponding to ρ0 and ρ1
are distinct in HQMpL; Zq Ď Hb2 pL; Zq.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

25

2.2. Hyperbolic Geometry. Throughout this paper, we reserve the letter K to denote R or C. We summarize some required facts from hyperbolic geometry of two
and three dimensions.
Maximal abelian subgroups
By identifying PSL2 pRq with the group of Möbius transformations on the upper
half-plane H2 of C, we have
PSL2 pRq – Isom` pH2 q ď Homeo` pS 1 q.
In particular, we have a restriction
rot : PSL2 pRq Ñ R{Z.
We also have
PSL2 pCq – Isom` pH3 q ď Homeo` pS 2 q.
A group µ ď PSL2 pKq is elementary if µ admits a finite orbit on Hd Y BHd for d “ 2
or 3.
For t P K, let us define the following elements in PSL2 pKq:
ˆ
˙
ˆ t{2
˙
ˆ
˙
cospt{2q sinpt{2q
e
0
1 t
Rotptq “
, expptq “
, pptq “
.
´ sinpt{2q cospt{2q
0 1
0 e´t{2
Let t P R. Then Rotptq is an elliptic rotation about i P H2 by the angle t. The
elements expptq and pptq are the hyperbolic translation by distance t along the geodesic r0, 8s and a parabolic isometry fixing 8, respectively. Each g P PSL2 pRq is
conjugate to one of these three types. In particular,
$
’
&|2 cospπ rotpgqq|, if g is elliptic;
(2.2.1)
| tr g| “ 2 coshp`pgq{2q,
if g is hyperbolic;
’
%2
if g is parabolic.
Here, `pgq denotes the translation length of g. We always regard the identity as an
elliptic, parabolic and hyperbolic isometry simultaneously.
Let us now suppose g P PSL2 pCq. Then g is conjugate to either expptq or pptq for
some t P C. We note that the following element is a rotation with the z–axis:
ˆ it{2
˙
e
0
exppitq “
.
0 e´it{2
Let C ď G be groups. We define the normalizer and the centralizer of C as
NG pCq “ NpCq “ tg P G : g´1Cg “ Cu,
ZG pCq “ ZpCq “ tg P G : rg, cs “ 1 for all c P Cu “

č
cPC

ZG pcq.

26

S. KIM, T. KOBERDA, AND M. MJ

Proposition 2.20.
(1) A maximal abelian subgroup of PSL2 pRq is conjugate to
one of the following groups:
SOp2q “ RotpRq,

exppRq,

ppRq.

(2) A maximal elementary subgroup of PSL2 pRq is conjugate to one of the following groups:
SOp2q “ RotpRq,

NpexppRqq “ exppRq ¸ Rotpπq,

ppRq.

(3) A maximal abelian subgroup of PSL2 pCq is conjugate to one of the following
groups:
exppCq,

ppCq.

(4) A maximal elementary subgroup of PSL2 pCq is conjugate to one of the following groups:
NpexppCqq “ exppCq ¸ Rotpπq,

ppCq.

For a maximal abelian subgroup µ of PSL2 pKq, we let Fix µ denote the set of
points x P Hd Y BHd which is fixed by all the elements in µ. The set Fix µ consists
of one or two points for d “ 2 or 3. The following are simple exercises in hyperbolic
geometry.
Lemma 2.21. Let µ be a maximal abelian subgroup of PSL2 pKq.
(1) For g P PSL2 pKq, the following are equivalent:
(i) Fix µ “ g Fix µ;
(ii) g P Npµq;
(iii) rc, gcg´1 s “ 1 for some c P µz1.
(2) For g P PSL2 pKq, we have Fix µ X g Fix µ ‰ ∅ if and only if one of the
following hold:
(i) µ is not hyperbolic and g P µ;
(ii) µ is hyperbolic and tr2 rc, gcg´1 s “ 4 for some c P µz1.
Finite type hyperbolic 2–orbifolds
A finite type 2–orbifold S is a geometric structure on an oriented surface of finite
topological type [99], which is prescribed by its signature
pg; m1 , m2 , . . . , mk q.
Here, g is the genus of S and the integers 2 ď m1 , . . . , mk ď 8 are the orders of
cone points for some k ě 0. The order mi “ 8 corresponds to a puncture or a
boundary. If S is a surface, then k “ 0.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

27

The Euler characteristic χpS q and the complexity ξpS q are then defined as:
˙
k ˆ
ÿ
1
χpS q “ 2 ´ 2g ´
1´
mi
i“1
ξpS q “ 3g ´ 3 ` k.
There is a complete hyperbolic structure on S if and only if χpS q ă 0.
A discrete subgroup L of PSL2 pRq is called a Fuchsian group. In this case, the
quotient space S “ H2 {L is a complete hyperbolic 2–orbifold and we write
L “ πorb
1 pS q.
If L is finitely generated, then S is of finite type. If, furthermore, L has a finite
co-volume, then we say L is a lattice. If S is compact, then L is said to be uniform.
Every finitely generated Fuchsian group is isomorphic to some lattice, since one
can contract a geodesic boundary of a complete hyperbolic 2–orbifold to a cusp
without changing its fundamental group.
Commutative–transitive groups
We say a group G is commutative–transitive if ra, bs “ ra, cs “ 1 implies rb, cs “ 1
for a, b, c P Gz1. The group Isom` pHn q is an example of a commutative–transitive
group. Every limit group is also commutative–transitive [8].
Lemma 2.22. Let G be a commutative–transitive group, and let C be a nontrivial
abelian subgroup of G.
(1) Then ZpCq is the unique maximal abelian subgroup of G containing C.
(2) If C is malnormal in G, then C is maximal abelian in G.
(3) If C is a maximal abelian subgroup of G, then we have
NpCq “ tg P G : rcg , cs “ 1 for all c P Cu
“ tg P G : rcg , cs “ 1 for some c P Cz1u
“ tg P G : C X C g ‰ 1u .
(4) Let A, µ be subgroups of G such that 1 ‰ C ď A X µ. Assume that µ is
maximal abelian in G and that C is malnormal in A. Then
C “ A X µ “ A X Npµq.
Proof. For part (1), let C ď G be a nontrivial abelian group and c P Cz1. Then
commutative–transitivity of G implies that the centralizer group ZpCq “ Zpcq is the
desired maximal abelian group containing C.
Part (2) is obvious. For part (3), let us write the given statement as
NpCq “ C1 “ C2 “ C3 .

28

S. KIM, T. KOBERDA, AND M. MJ

Then the inclusions
NpCq Ď C1 Ď C2 Ď C3
are immediate by definition and the maximality. To see C3 Ď NpCq, let us suppose
C X C g ‰ 1 for some g P G. We can find c0 , c1 P Cz1 such that c0 “ cg1 . Then
C “ Zpc1 q “ Zpc0 qg “ C g
and hence, g P NpCq.
For part (4), let g P A X Npµq. Since C g ď µg “ µ, for each c P C we have
rc, cg s “ 1. By part (2), the group C is maximal abelian in A. So, part (3) implies
that
g P NA pCq “ C,
as desired.

Minimalization of Fuchsian groups
The following characterizes minimalization of Fuchsian groups.
Lemma 2.23. Suppose we have a nonelementary Fuchsian representation
ρ0 : L Ñ PSL2 pRq
for some finitely generated torsion-free group L such that Λpρ0 q ‰ S 1 . Then there
exists a path tρt u of Fuchsian representations of L such that Λpρ1 q “ S 1 and such
that ρ1 pLq contains a nontrivial parabolic element. Furthermore, ρ1 is a minimalization of ρ0 .
Proof. Let X denote the space of the Fuchsian representations of L. Put S 0 “
H2 {ρ0 pLq. The surface S 0 contains at least one geodesic funnel, represented by say,
h P L.
There exists a continuous deformation tS t u of the hyperbolic surface S 0 to a
finite-area hyperbolic surface S 1 . This is done by deforming all the geodesic funnels
to cusps; the details of this construction are worked out by Floyd in [41].
Let tρt u Ď X be a path corresponding to tS t u. Then ρ1 phq must be a nontrivial
parabolic element. Part (6) of Theorem 2.12 implies that ρ1 is semi-conjugate to ρ0 .
Hence, ρ1 is a minimalization of ρ0 .

2.3. Indiscrete Subgroups of PSL2 pRq. It will be useful for us to have a dynamical characterization of subgroups of PSL2 pRq which are not discrete. The following
lemma is well-known to experts in algebraic groups and Lie groups; see [2, Theorem 8.4.1], [61] and [92] for instance. We include a proof for the convenience of
the reader.
Lemma 2.24. For a finitely generated non-elementary subgroup L of PSL2 pRq, the
following are equivalent:
(1) The rotation spectrum of L is infinite;

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

29

(2) Some element of L is an elliptic element of infinite order;
(3) The group L is indiscrete;
(4) The group L is dense.
Proof. Note that L is Zariski dense, since a subgroup of PSL2 pRq which is not
virtually solvable cannot be contained in a Borel subgroup and hence is Zariski
dense.
We note L is dense if and only if it is Zariski dense and indiscrete. Indeed, if
L is dense then its Zariski closure is a closed subgroup of PSL2 pRq containing L,
and hence is all of PSL2 pRq. Conversely, suppose L is Zariski dense and indiscrete.
Then the closure of L in the usual topology on PSL2 pRq contains a component
of positive dimension which is a closed subgroup and is hence a Lie subgroup of
PSL2 pRq. Since L is Zariski dense, this subgroup must be all of PSL2 pRq. Thus, we
establish the equivalence of (3) and (4).
Since the density of L ď PSL2 pRq implies the density of tr2 pLq Ď R, we see the
implication (4)ñ p1q.
To see that (1) implies (2), suppose L ď PSL2 pRq has infinite rotation spectrum
but L does contain an irrational rotation. Then L must contain elliptic elements of
arbitrarily high order. Moreover, being a finitely generated linear group, we have
that L contains a finite index subgroup H which is torsion-free by Selberg’s Lemma.
If n is the index of H in L and g P L is arbitrary, then gn! P H. It follows that L must
have a bounded exponent, a contradiction.
The implication (2)ñ(3) is obvious.

We remark that Lemma 2.24 really is specific to PSL2 pRq. It fails already in
PSL2 pCq, where “infinite rotation spectrum” is replaced by “infinite order elliptic
element” (see [70]).
To illustrate Lemma 2.24 a little, we give an example of a dense subgroup of
PSL2 pRq with infinite purely arithmetic rotation spectrum,
and we explicitly find
?
elliptic elements. Consider the ring of integers Zr 2s. For an arbitrary  ą 0,
choose elements
?
a, b, c, d P Zr 2s
such that
?
(i) 0 ă a ` d ă  (this is possible as Zr 2s is dense in R),
(ii) b “ 1, c “ ad ´ 1.
The matrix
ˆ
˙
a b
P PSL2 pRq
c d
is easily checked to lie in the finitely generated dense subgroup
?
PSL2 pZr 2sq ă PSL2 pRq,

30

S. KIM, T. KOBERDA, AND M. MJ

and to be an elliptic element of PSL2 pRq. The rotation number θ is given by
2 cos

θ
“ a ` d,
2

which in turn lies in the interval
r2 arccosp{2q, πs,
thus providing infinitely many distinct arithmetic rotation numbers. More generally,
such examples can be constructed from Hilbert modular groups PSL2 pOq, where O
is the ring of integers in a totally real number field.
The following observation is an immediate consequence of Lemma 2.24:
Corollary 2.25. If L ď PSL2 pRq is a finitely generated group such that rotpLq “
t0u, then L is torsion-free and discrete.
3. Topological Baumslag Lemma
This section deals with one of the principal technical tools of the paper, namely
an extension of Baumslag Lemma. The following are some of the main ingredients
in this section:
(1) A Topological Baumslag Lemma, which gives sufficient conditions to guarantee nontriviality of a word
wpt1 , ¨ ¨ ¨ , tk q “ g1 µ1 pt1 q ¨ ¨ ¨ gk µk ptk q
for large values of the parameters ti . Here the µ j pt j qs are one-parameter subgroups of a continuous (possibly analytic) group. The proof is quite general
and reminiscent of Tits’ proof [100] of the Tits’ alternative for discrete linear groups in that the underlying idea consists of a ping-pong argument.
(2) The one-parameter subgroups are often (generalizations of) parabolic and
hyperbolic one-parameter subgroups of PSL2 pRq. To handle elliptic subgroups we use a complexification and Zariski density trick by embedding
PSL2 pRq in PSL2 pCq and reduce the elliptic case to the hyperbolic one
(Lemma 3.7).
We focus primarily on projective group actions on the circle, i.e. representations
into PSL2 pRq; but many of the proofs go through in the more general context of
representations into linear algebraic semi-simple groups, including PSL2 pCq.
3.1. Topological Setting. We will often denote a generator of Z by s. Recall our
notation from the introduction:
A˚C “ A ˚C pC ˆ xsyq.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

31

Let Fn denote a free group of rank n. Half a century ago, Baumslag used the
following lemma to deduce the residual freeness of groups of the form Fn ˚xwy Fn
for w P Fn .
Lemma 3.1 ([6]). Suppose we have
u, g1 , g2 , . . . , gk P Fn
such that ru, gi s ‰ 1 for each i. Then there exists M ą 0 such that
g1 ut1 g2 ut2 ¨ ¨ ¨ gk utk ‰ 1
whenever |ti | ě M for each i.
Sometimes called Baumslag Lemma, this result plays a fundamental role in the
theory of limit groups; see [102] and references therein. Baumslag Lemma generalizes for multiple “twisting words” [64, 5], and also for torsion-free word-hyperbolic
groups [51]. We will present a continuous version of Baumslag lemma in this section that will imply both of these generalizations.
Throughout this subsection, we let T be a set (of parameters) equipped with a
decreasing sequence of subsets
T Ě T0 Ě T1 Ě T2 Ě ¨ ¨ ¨ .
Definition 3.2. Let X be a topological space and let φ : T Ñ HomeopXq be a map.
(1) We say φ is attracting (to C) if we have a nonempty proper compact set
C Ď X such that for each open U Ě C there exists M ą 0 satisfying
φptq˘1 pXzUq Ď U for all t P T M .
(2) We say φ is doubly attracting (to pC ` , C ´ q) if we have a disjoint pair of
nonempty compact sets pC ` , C ´ q in X such that X ‰ C ` Y C ´ and such
that for each disjoint pair of open neighborhoods pU ` , U ´ q of pC ` , C ´ q
there exists M ą 0 satisfying the following for all t P T M :
φptqpXzU ´ q Ď U ` ,

φptq´1 pXzU ` q Ď U ´ .

Remark 3.3.
(1) We do not assume φ is continuous or a homomorphism.
(2) We forbid the possibility of X “ C ` Y C ´ in (2), since we do not want to
call the constant map
φ : T Ñ tIdu
as doubly attracting.
(3) A doubly attracting map is attracting: we simply set C “ C ` Y C ´ .
Example 3.4. Let us list some examples of attracting maps. We let K “ R or C.

32

S. KIM, T. KOBERDA, AND M. MJ

(1) Suppose µ is a maximal parabolic subgroup of PSL2 pKq. Up to conjugacy,
we can parametrize µ as
ˆ
˙
1 z
µpzq “
.
0 1
Set T “ K and T M “ tz P T : |z| ě Mu. Then for d “ 2 or 3, the map
µ : T Ñ PSL2 pKq – Isom` pHd q ď Homeo` pS d´1 q
is attracting to Fix µ.
(2) A maximal hyperbolic subgroup µ ď PSL2 pKq can be parametrized as
ˆ
˙
z 0
µpzq “
0 1{z
up to conjugacy. If we let T “ Kˆ and T M “ tz P T : |z| ě Mu, then
µ : T Ñ PSL2 pKq ď Homeo` pS d´1 q
is again attracting to Fix µ.
(3) If µ is a maximal elementary subgroup of Isom` pHn q for some n ě 3, then
the natural embedding φ : T “ µ ãÑ Homeo` pS n´1 q is attracting to Fix µ,
for a suitable stratification T “ T 0 Ě T 1 Ě ¨ ¨ ¨ .
(4) For 1 ď k ď 8, we have a k–fold cover:
pkq

PSL2 pRq Ñ PSL2 pRq,
Let µ ď PSL2 pRq be a maximal non-elliptic abelian subgroup and µ„ ď
PSLpkq p2, Rq be the lift of µ. Using the parametrization in (1) and (2), we
see
φ : T “ µ„ ãÑ Homeo` pS 1 q
is attracting to the periodic set Per µ„ .
(5) Let g P Isom` pHn q and define φ : T “ Z Ñ Homeo` pS n´1 q as φptq “ gt .
We set T M “ tt P Z : |t| ě Mu. Then φ is attracting to Fix g if and only if g
is non-elliptic.
(6) Let S be a closed hyperbolic surface and let g P ModpS q be pseudo-Anosov.
We have an action of ModpS q on BTeichpS q “ PMLpS q, and also on the
boundary of the curve graph of S . Then the map φ : t ÞÑ gt is attracting to
the union of the stable and the unstable laminations. In addition to these
actions, we have an action of ModpS , xq on S 1 “ BH2 such that the map
t ÞÑ gt is attracting to the set consisting of even number of points [29].
Here, x denotes a base point of S .

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

33

(7) Let G be a torsion-free non-elementary word-hyperbolic group acting on BG
by homeomorphisms, and let 1 ‰ h P G. Then the map φ : t ÞÑ ht is attracting to th8 , h´8 u. Note that BG ‰ th8 , h´8 u, as required by Definition 3.2
(2).
Lemma 3.5.
(1) If φ : T Ñ HomeopXq is doubly attracting for some Hausdorff
space X, then for all sufficiently large M and for each t P T M , the group
xφptqy is an infinite cyclic subgroup of HomeopXq.
(2) Let φ : T Ñ Isom` pHd q ď Homeo` pS d´1 q be doubly attracting for some
d ě 2. Then for all sufficiently large M and for each t P T M , the element
φptq is hyperbolic.
Proof. (1) Let φ doubly attract to pC ` , C ´ q. Since X ‰ C ` Y C ´ by assumption,
we can find disjoint open neighborhoods U ˘ of C ˘ and
x0 R U ` Y U ´ .
If M is sufficiently large and t P T M , then
φptq˘1 .x0 P U ˘ Ď XzU ¯ .
So for each n ą 0, we have
In particular, φptq˘n

φptq˘n .x0 P U ˘ .
‰ Id. Now (2) is immediate.



The following is the main result of this section, and called as a Topological
Baumslag Lemma. Some of the ideas go back to the ping-pong arguments of
Klein [66], Maskit [76] and Tits [100].
Lemma 3.6 (Topological Baumslag Lemma). Let X be a Hausdorff space and let
k ě 1. Define ψ : T Ñ HomeopXq by
ψptq “ g1 φ1 ptqg2 φ2 ptq ¨ ¨ ¨ gk φk ptq,
such that the following hold for each i “ 1, 2, . . . , k:
(i) gi P HomeopXq;
(ii) φi : T Ñ HomeopXq is attracting to Ci ;
(iii) Ci X gi`1Ci`1 “ ∅ where indices are taken modulo k;
(iv) X ‰ g1C1 Y Ck .
Then ψ is doubly attracting to pg1C1 , Ck q.
Proof. Choose an open neighborhood Ui of Ci such that
Ui X gi`1 pUi`1 q “ ∅
for each 1 ď i ď k. There exists M ą 0 such that
φi pT M q˘1 pXzUi q Ď Ui .

34

S. KIM, T. KOBERDA, AND M. MJ

Fix t such that t P T M . It suffices to verify the ping-pong condition of Definition 3.2
(2) for ψptq with respect to the pair of disjoint open sets pg1 pU1 q, Uk q. We first note
gi φi ptqpXzUi q Ď gi pUi q Ď XzUi´1 ,
´1
φi ptq´1 g´1
i pXzgi pU i qq “ φi ptq pXzU i q Ď U i Ď Xzgi`1 pU i`1 q.
So we have that
k
k´1
ź
ź
ψptqpXzUk q “
gi φi ptqpXzUk q Ď
gi φi ptqpXzUk´1 q
i“1

i“1

Ď ¨ ¨ ¨ Ď g1 φ1 ptqpXzU1 q Ď g1 pU1 q,
ψptq pXzg1 pU1 qq “
´1

1
ź

φi ptq´1 g´1
i pXzg1 pU 1 qq

2
ź

Ď

i“k

φi ptq´1 g´1
i pXzg2 pU 2 qq

i“k

Ď ¨¨¨ Ď

φk ptq´1 g´1
k pXzgk pU k qq

“ φk ptq´1 pXzUk q Ď Uk .


Thus we establish the lemma.

3.2. Projective and Discrete Settings. We will mainly employ Lemma 3.6 for the
projective setting. For a subgroup H ď G, we let NpHq denote the normalizer
group of H. For each matrix g, we let }g} denote its `2 -norm. Recall our notation
that K “ C or K “ R.
Lemma 3.7 (Projective Baumslag Lemma). Let µ ď PSL2 pKq be a maximal abelian
subgroup. We define φ : µ Ñ PSL2 pKq by
φpνq “ g1 νm1 g2 νm2 ¨ ¨ ¨ gk νmk ,
such that k ě 1 and such that for each i “ 1, . . . , k we have
(i) gi P PSL2 pKq and mi P Zzt0u;
(ii) Fix µ X gi Fix µ “ ∅.
(1) Then the following set is finite for each x P K:
tν P µ : tr2 ˝φpνq “ xu.
(2) If µ is not purely elliptic, then we have
lim

| tr ˝φpνq| “ 8.

}ν}Ñ8,νPµ

Proof. Let us first consider the case K “ C. We will prove (2) and then (1). Since
the condition }ν} Ñ 8 is invariant under conjugation, we may parametrize µ so
that
ˆ
˙
ˆ
˙
1 z
z 0
.
µpzq “
or µpzq “
0 1
0 1{z

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

35

In order to apply Topological Baumslag Lemma (Lemma 3.1), we set
φi pzq “ µpmi zq,

Ci “ Fix µ Ď S 2 ,

ψpzq “ φ ˝ µpzq.

Note gi Fix µ X Fix µ “ ∅ on S 2 . So ψ is doubly attracting to pg1 Fix µ, Fix µq. As
the hyperbolic distance between g1 Fix µ and Fix µ is infinite, we have
lim | tr2 ˝ψpzq| “ lim | tr2 ˝φ ˝ µpzq| “ 8.
|z|Ñ8

|z|Ñ8

So part (2) follows. For part (1) we note that tr2 ˝φ ˝ µpzq “ x is an algebraic
equation which is not an identity, and hence, the solution set is finite.
Now we set K “ R. If µ is non-elliptic, then the proofs of (1) and (2) are almost
identical to the previous paragraph. So we assume µ is elliptic and prove (1).
We use a complexification trick, embedding PSL2 pRq into PSL2 pCq and using
Zariski density of SOp2q in Cˆ . That is, we regard H2 as a hemisphere properly
embedded in H3 with the upper half-space model. We may assume µ is the rotation
group about the z–axis of H3 , so that
ˆ it{2
˙
e
0
µptq “
and φ ˝ µp2tq “ ψpeit q,
0 e´it
where we define
ψpzq “

k
ź
i“1

ˆ
gi

z 0
0 z´1

˙mi
.

By assumption, we have
gi .t0, 8u X t0, 8u “ ∅.
We define an algebraic set
Y “ tz P Cˆ : tr2 ˝ψpzq “ xu.
If some Y X S 1 is infinite, then Y “ Cˆ since every infinite set is Zariski dense in
Cˆ . This is a contradiction to the case K “ C.

Remark 3.8. Let us illustrate why one cannot weaken the condition (ii) to
(ii)’ Fix µ ‰ gi Fix µ
in the above lemma. Consider µptq “ expptq and g1 , g2 are two distinct nontrivial
parabolic elements fixing 8. In the upper half-space model, we see
rµptq, gi µptqg´1
i s
is a translation along R, and so
“
‰
´1
f ptq “ rµptq, g1 µptqg´1
s,
rµptq,
g
µptqg
s
2
1
2
is constantly the identity. Note that f ptq satisfies the condition (ii)’ and also all the
conditions of the lemma except for (ii).

36

S. KIM, T. KOBERDA, AND M. MJ

A version of Baumslag Lemma holds for other groups with north-south dynamics. Recall an isometric action of a non-virtually cyclic group G on a metric space
pX, dq is acylindrical if for all r ě 0 there exists R, N ě 0 such that whenever
x, y P X satisfy dpx, yq ě R the set
tg P G : dpx, gxq ď r and dpy, gyq ď ru
has cardinality at most N. If G acts acylindrically on a hyperbolic space X, then
each element g P G is either elliptic (that is, xgy has a bounded orbit) or loxodromic (that is, g acts by translation on a quasi-geodesic axis); see [13] and [52].
Word-hyperbolic groups, mapping class groups and right-angled Artin groups admit acylindrical actions on hyperbolic spaces [13, 65, 32]; see also [9]. Using
a local-to-global principle, we can strengthen Topological Baumslag Lemma for
acylindrically hyperbolic groups to draw a geometric conclusion.
Theorem 3.9 (Discrete Baumslag Lemma). Let G be a finitely generated group
acting acylindrically on a hyperbolic space Y and let
g1 , . . . , gk , u1 , . . . , uk P G
such that each ui is loxodromic. For each 1 ď i ď k, we assume
xui y X gi`1 xui`1 yg´1
i`1 “ t1u,
where the indices are taken modulo k. For t “ pt1 , . . . , tk q P Zk , we define
φptq “ g1 ut11 g2 ut22 ¨ ¨ ¨ gk utkk .
Then φptq is loxodromic whenever all |ti |’s are sufficiently large.
Proof. The Gromov boundary BY is a complete metric space [50, 1.8B]. Each loxodromic u P G admits a quasi-geodesic axis `u Ď Y, and acts by north-south dynamics on BY with exactly two fixed points which comprise B`u [52]. For Epuq “
StabpB`u q, we have rEpuq : xuys ă 8. See [9, Proposition 6] and [32, Lemma 6.5
and Theorem 6.8] for relevant details. So for each 1 ď i ď k, we have
Epui q ‰ gi`1 Epui`1 qg´1
i`1 ,

B`ui X gi`1 B`ui`1 “ ∅.

Let us fix y P Y. For each t P Zk , we naturally regard φptq as a word of length
k ` |t1 | ` ¨ ¨ ¨ ` |tk |
and define a bi-infinite coarse path
γ: Z Ñ Y
which is the orbit of the bi-infinite words ¨ ¨ ¨ φptqφptq ¨ ¨ ¨ based at y.
It follows from [17, III.H.1.13] or [43, Theorem 21, Chapter 5] that for each
` ě 1, there exists (large) K such that every pK, `q-local quasi-geodesic in Y is a

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

37

global quasi-geodesic. We shall show that γ is such a pK, `q-local quasi-geodesic
whenever each |ti | is sufficiently large; hence φptq is loxodromic.
It remains to choose each |ti | large enough so that γ is a pK, `q-local quasigeodesic with appropriate K, `. First note that
(1) There exists `1 such that for all i and n P Z, the word uni is an p`1 , `1 q–quasiisometric embedding of r0, |n|s. This follows from the hypothesis that ui ’s
are loxodromic.
2
(2) There exists `2 such that for all i and n1 , n2 P Z, the word uni 1 gi`1 uni`1
is
an p`2 , `2 q–quasi-isometric embedding of r0, |n1 | ` |n2 | ` 1s. This follows
from Item (1) above and the hypothesis that B`ui X gi`1 B`ui`1 “ ∅ (see for
instance Corollary 1.37 of [84]).
Choose ` “ maxp`1 , `2 q and let K be large enough so that every pK, `q-local
quasi-geodesic is a global quasi-geodesic. We now choose each |ti | ě K. Then
each subpath γpN, N ` Kq of γ is of type (1) or (2) above and is therefore an p`, `qquasi-geodesic. Hence γ is a global quasigeodesic and we are done.

Remark 3.10. In the above proof, we can apply Lemmas 3.5 and 3.6 for the setting
T “Z,
k

X “ BY,

φi ptq “

utii

φptq “

k
ź

gi φi ptq,

i“1

to conclude that φptq ‰ 1. However, this does not show that φptq is loxodromic.
Remark 3.11. A. Sisto told us a variation of Theorem 3.9 holds for the case when
each ui to be an infinite elliptic element [95].
Example 3.12. Suppose G is a torsion-free word-hyperbolic group. Define φ : Zk Ñ
G as in Theorem 3.9, where gi , ui P G satisfy
rui , gi`1 ui`1 g´1
i`1 s ‰ 1.
Then φptq ‰ 1 whenever all |ti |’s are sufficiently large by Theorem 3.9. If
u1 “ ¨ ¨ ¨ “ uk “ u,
the hypothesis can be weakened to
ru, gi s ‰ 1
for each i. So we have a direct generalization of the original Baumslag’s Lemma.
See also [51, Lemma 5.4], [64, Lemma 3.6] and [5, Lemma 3.5] for other special
cases of Theorem 3.9.
Corollary 3.13. Let S be a closed oriented hyperbolic surface and ModpS q denote
its mapping class group. Define φ : Zk Ñ ModpS q as in Theorem 3.9, where gi , ui P
G satisfy that
rui , gi`1 ui`1 g´1
i`1 s ‰ 1,

38

S. KIM, T. KOBERDA, AND M. MJ

and that each ui is pseudo-Anosov. Then φptq is pseudo-Anosov whenever all |ti |’s
are sufficiently large.
Proof. Since G acts on the curve graph acylindrically [13], we can apply Theorem 3.9.

Remark 3.14. If u1 “ ¨ ¨ ¨ “ uk in the above corollary, then we can also weaken the
hypothesis to
ru, gi s ‰ 1
for each i, since mapping class groups do not admit Baumslag–Solitar relations that
are not commutators [62, Theorem 8.2].
The following is an immediate consequence of Corollary 3.13.
Proposition 3.15. Let G be the mapping class group of a closed orientable hyperbolic surface, and let x f y ď G be a maximal cyclic pseudo-Anosov subgroup.
Define φn : G ˚x f y G Ñ G by extending the map IdG on the left factor and the map
g ÞÑ f n g f ´n
on the right factor. Then the family tφn u is eventually injective; that is, we have
č ď
ker φn “ t1u.
Mě1 něM

4. Splittable Fuchsian Groups
In this section, we study deformations of (possibly indiscrete) faithful representations of Fuchsian groups such that almost all points on the deformations are still
faithful. Let L be a splittable Fuchsian group, which includes all Fuchsian groups
with Euler characteristic at most ´1; see Definition 4.8. We will prove that an arbitrarily small deformation of a given representation can be chosen so that the new
trace spectrum is almost disjoint from the original one (Theorem 4.11). Then we
show Xproj pLq contains at least one indiscrete representation (Lemma 4.14). Moreover, if an open set U contains at least one indiscrete representation in Xproj pLq,
then U contains uncountably many pairwise inequivalent indiscrete representation
in Xproj pLq (Theorem 4.16).
The main engine behind this study is the Pulling-Apart Lemma (Lemmas 4.6
and 4.7). Roughly, this lemma means that “a generic deformation of a representation remembers obvious group relations only”. The two ingredients of this lemma
are put together here. First, we use in an essential way, the algebro-geometric structure of representation varieties to establish that a generic point is faithful provided
no word (varying according to suitably chosen parameters) is identically trivial.
Topological Baumslag Lemma and the complexification trick from the previous
section furnish this sufficient condition. While building faithful representations of

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

39

free products with amalgamation G “ A ˚C B, or HNN extensions A˚φ from faithful
representations of A, B, we will use genericity or Baire category argument, which
is the second ingredient. This argument makes precise the notion of a property
satisfied by a generic point in a real or complex algebraic variety.
Throughout this section, we will let K “ C or K “ R.
4.1. Very General Points, Abundance and Stable Injectivity. Our techniques
involve both the topological category as well as that of (real or complex) algebraic
varieties. Thus a choice of terminology needs to be made, which we set out to do
here.
First we set up the terminology in the topological category. Recall that a Gδ set in
a topological space X is the countable intersection of open subsets. We shall refer
to a property as being satisfied by a very general point in X if it is satisfied by a
point in a (suitably chosen) dense Gδ subset of X.
The corresponding notion in algebraic geometry is more restrictive. We will regard linear algebraic groups as affine algebraic sets and equip them with the Zariski
topology. We shall refer to a property as being satisfied by a very general point in
an algebraic set X if it is satisfied by a point in the complement of a countable union
of subsets
tX1 , X2 , . . .u
of X such that each Xi is a proper algebraic subset of an irreducible component of
X.
We gather together some facts for use later. All algebraic sets will be defined
over R or C. Note that PSL2 pRq and PSL2 pCq are irreducible algebraic groups. The
following is a standard fact; see [15, Lemma 2.4] for instance.
Lemma 4.1. Let X be an algebraic set, and let
X1 Ď X2 Ď ¨ ¨ ¨
be a countable chain of proper algebraic subsets of X. Then the set
8
ď
D “ Xz Xi
i“1

contains very general points of some irreducible component of X. Furthermore, if
dim X ą 0, then D is uncountable.
Proof. Each Xi is closed in the Zariski topology, and hence is either nowhere dense
in X or contains an irreducible component of X. Since X has finitely many components and each Xi is a proper subset, there exists an irreducible component V of X
such that V X Xi is proper in V for each i. By definition,
ď
Y “ Vz Xi
i

40

S. KIM, T. KOBERDA, AND M. MJ

is very general in V. For the second part, note Y contains a dense Gδ set.



We indicate here a general method of extracting a large (uncountable) family of
faithful representations by applying Lemma 4.1. The main idea in this comes from
translating a concept from the (discrete) context of limit groups into the (algebraic
geometry) context of representations into algebraic groups.
Definition 4.2 ([93]). We say a sequence of group homomorphisms
p fn qně1 Ď HompG, Hn q
is stably injective if for each g P Gzt1u there exists n0 ą 0 such that fn pgq ‰ 1 for
all n ě n0 .
For terminological convenience, we will also say a sequence of maps is stably
injective if the sequence has a stably injective subsequence. As a warm-up, we now
give a simple illustration of the Baire category argument in connection with stably
injective maps.
Lemma 4.3. Let L be a finitely generated group, and G be an algebraic group. If
there exists a stably injective sequence of representations
t fn : L Ñ GunPN
which contains infinitely many distinct representations, then faithful representations
are very general in some irreducible component of HompL, Gq.
Proof. Write L “ xS y. Note that
X “ HompL, Gq Ď GS
is an algebraic set (by the Hilbert Basis Theorem). Since X is infinite, it has a
positive dimension.
For each finite subset Q Ď Lz1, we define an algebraic set
YpQq “ tρ P X : ρpgq “ 1 for some g P Qu.
The stable injectivity implies that YpQq is proper for all n P N. We can now apply
Lemma 4.1 to the set of faithful representations, which can be written as
ď
D “ Xz
YpQq.

QĎL, finite

Definition 4.4. A group L is a limit group if there exists a stably injective sequence
tρn : L Ñ FunPN
for some fixed nonabelian free group F.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

41

Limit groups are torsion-free and finitely presented [8]. Surface groups and free
groups are limit groups. An obvious consequence of Lemma 4.3 is that if an algebraic group G contains a nonabelian free group F, then G contains an isomorphic
copy of every limit group; see [8, 102, 5].
The following group–theoretic lemma will come handy when finding stably injective sequences. For a group L, a subset P Ď L and an integer n ą 0, we denote
Pn “ tgn : g P Pu.
Lemma 4.5. For a residually finite group L and a finite subset P Ď L, the following
sequence of the quotient maps is stably injective:
tρn : L Ñ L{xxPn! yy : n ě 1u.
Proof. If N is a normal subgroup of L with index n, then xxPn! yy ď N. So we have
č
xxPn! yy “ t1u.

ně1

4.2. Pulling-Apart Lemma. Let us now describe a more general method to continuously “pull-apart” subgroups to become (amalgamated) free products of those
subgroups. Similar ideas can be found in the literature, e.g. [8] and [73]. McCammond and Sulway coined the term “pull-apart” in a different (but loosely relevant)
sense from ours; see [79].
Recall our convention that K “ R or K “ C. For a group C ď PSL2 pKq, we let
NpCq denote its normalizer group in PSL2 pKq. Suppose C ď PSL2 pKq is nontrivial
and abelian, so that the centralizer ZpCq is the maximal abelian group containing
C; see Lemma 2.22. If C is hyperbolic, then rNpCq : ZpCqs “ 2. Otherwise, we
have NpCq “ ZpCq.
Using the multiplicative notation, we let xsy – Z. For g, t P PSL2 pKq, we define
homomorphisms Innptq : PSL2 pKq Ñ PSL2 pKq and δt : Z Ñ PSL2 pKq by
Innptqpgq “ gt ,

δt psq “ t.

If L is a group and V Ď L is a set, we define
V L “ tvg : v P V, g P Lu.
The following can be regarded as a warm-up case for the Amalgamated PullingApart Lemma.
Lemma 4.6 (Free Product Pulling-Apart Lemma).
(1) Let A and B be countable subgroups of PSL2 pKq, and let W Ď K be a countable set. Then for a
very general ν P PSL2 pKq, the group Lν “ xA, Bν y is isomorphic to A ˚ B
and furthermore,
`
˘
tr2 Lν zpA Y Bν qLν X W “ ∅.

42

S. KIM, T. KOBERDA, AND M. MJ

(2) Let A be a countable subgroup of PSL2 pKq, and let W Ď K be a countable set. Then for a very general ν P PSL2 pKq, the group Lν “ xA, νy is
isomorphic to A ˚ Z and furthermore,
`
˘
tr2 Lν zALν X W “ ∅.
Proof. In Case (1), we let
L “ A ˚ B,

V “ A Y B Ď L.

In Case (2), we let
L “ A ˚ xsy, V “ A Ď L
for xsy – Z. We have natural maps (more precisely defined below)
ρν : L Ñ Lν .
Using Lemma 2.21, we can find a hyperbolic maximal abelian group µ ď PSL2 pKq
such that for all h P xVyzt1u, we have
h Fix µ X Fix µ “ ∅.
L

For T Ď K and Q Ď LzV , we define
C pQ, T q :“ tν P PSL2 pKq : tr2 ˝ρν pQq X T ‰ ∅u.
The crucial step of the proof is the following claim.
Claim. If T Ď K and Q Ď LzV L are finite subsets, then C pQ, T q ‰ PSL2 pKq.
Let us prove the claim in the cases (1) and (2) separately.
(1) For each ν P PSL2 pKq, we define ρν by the following diagram:
@A
IdA

0

$

1n

ρν



L “: A ˚ B

/

+
3

PSL2 pKq

Innpνq˝IdB

B

Each g P Q can be written as
g“

k
ź

ai bi

i“1

for some k ě 1, ai P Az1 and bi P Bz1 up to conjugacy. It follows that
ρν pgq “

`
ź
i“1

gi νmi

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

43

for some gi P pA Y Bqz1 and mi “ ˘1. By Lemma 2.21 and Projective Baumslag
Lemma (Lemma 3.7), whenever ν P µ and }ν} " 0 we have
tr2 ˝ρν pQq X T “ ∅.
(2) For each ν P PSL2 pKq, we define ρν as
?

A
IdA

/

%

1o

ρν

L “ A ˚ xsy
9

/
3

+

PSL2 pKq

sÞÑν



xsy
Let g P Q. If g “ sm for some m ‰ 0 (up to conjugacy), then we have
ρν pgq “ νm .
Otherwise, we have
ρν pgq “

`
ź

gi νmi

i“1

for some k ě 1, gi P Az1 and mi P Zz0 up to conjugacy. The rest of the proof is
identical to (1).
From the claim above and Lemma 4.1, we see that the following set is very
general:
PSL2 pKqzC pLzV L , W Y t4uq.
Note that if 1 ‰ g P V L , then ρν pgq ‰ 1. Hence we can further require that ρν is
faithful.

For C ď PSL2 pKq, recall our notation for the centralizer:
ZpCq “ tg P PSL2 pKq : rg, cs “ 1 for all c P Cu.
Lemma 4.7 (Amalgamted Pulling-Apart Lemma). For a countable subgroup L ď
PSL2 pKq and a countable set W Ď K, assume one of the following.
(1) For some subgroups A, B, C ď L, we have that L “ xA, By and that C
is a malnormal nontrivial abelian subgroup of A and also of B. For each

44

S. KIM, T. KOBERDA, AND M. MJ

ν P ZpCq we let ρν be the map uniquely determined by the following commutative diagram
@A
IdA

0

%

Cn

ρν

L˚ “
A ˚C B
9


3

+
/

PSL2 pKq

Innpνq˝IdB

B
We further assume V “ A Y B is parabolic-free.
(2) For some subgroups A, C ď L and an element s P L, we have that L “
xA, sy and that C and C s are malnormal nontrivial abelian subgroups of A.
Furthermore, we assume that either
‚ s P ZpCq, or
‚ C and C s are not conjugate in A.
For each ν P ZpCq we let ρν be the map uniquely determined by the following commutative diagram
9

A
IdA

+

(

C ˚ Cs

L˚ “ A˚Innpsq : CÑC s

ρν

6

%

/,2

PSL2 pKq

s˚ ÞÑνs

C ˚ xs˚ y
Here s˚ denotes the stable generator of L˚ . We further assume V “ A is
parabolic-free.
Under the assumption (1) or (2), for all but countably many ν P ZpCq the map ρν is
faithful and
´
¯
˚
tr2 ˝ρν L˚ zV L X W “ ∅.
Proof. We let µ “ ZpCq, which is a non-parabolic maximal abelian group. Also,
put V “ A Y B in (1), and V “ A in (2), as subsets of L˚ . As in the proof of
Lemma 4.6, it suffices to show the following claim.
˚

Claim (Key Claim). If T Ď K and Q Ď L˚ zV L are finite subsets, then we have
C pQ, T q :“ tν P µ : tr2 ˝ρν pQq X T ‰ ∅u ‰ µ.
From the Key Claim, we see that the following set is very general, as desired:
´
¯
˚
µzC L˚ zV L , t4u Y W .

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

45

Let us now prove the Key Claim through a sequence of steps.
Claim 1. (i) We have Npµq X V “ C ď µ, as subsets of PSL2 pKq.
(ii) For h P VzC, we have Fix µ X h Fix µ “ ∅.
(iii) In Case (2), if s R ZpCq, then we have Fix µ X s Fix µ “ ∅.
If C is not hyperbolic in (i), then Npµq X V “ µ X V “ C by Proposition 2.20. So
we may assume C is hyperbolic and in particular, torsion-free. Then for each c P C
and h P Npµq X V, we have hch´1 “ c˘1 . Hence h normalizes C and this proves
part (i) of Claim 1. For part (ii), let us fix c P Cz1. Note that
tr2 rc, hch´1 s ‰ 4
by part (i) and our hypothesis that V is parabolic-free. By Lemma 2.21 the proof
of part (ii) is complete. For part (iii), let us fix c P Cz1. Since C s ‰ C, we see
s R Npµq and hence,
rc, scs´1 s ‰ 1.
As A is parabolic-free and xC, C s y ď A, we have
tr2 rc, scs´1 s ‰ 4.
This completes the proof of Claim 1.
Let us now consider the cases (1) and (2) of the lemma separately.
Case (1) L˚ “ A ˚C B.
Let g P Q. Up to conjugacy, we can write
ρν pgq “

k
ź

gi νmi

i“1

for some k ě 1, mi “ ˘1 and
gi P A Y BzC “ A Y BzNpµq.
Lemma 2.21 implies that
rc, gi cg´1
i s P A Y Bz1
for all i and c P Cz1. If Fix µ X gi Fix µ ‰ ∅, then µ must be hyperbolic and
tr2 rc, gi cg´1
i s “ 4.
This contradicts the parabolic-free hypothesis. Lemma 3.7 implies the Key Claim.
Case (2)-1. L˚ “ A˚C and C ď Zpsq.
In this case, we have
s P ppLzAq Y Cq X µ.
If g “ cps˚ qm P Q for some c P C and m P Zz0, then we have
tr2 ˝ρν pgq “ tr2 pcνm sm q P tr2 pµq
for ν P µ. Then tr2 ˝ρν pgq R T whenever }ν} " 0.

46

S. KIM, T. KOBERDA, AND M. MJ

So up to conjugacy, we consider
g“

k
ź

gi ps˚ qmi P Q

i“1

for some k ě 1, some gi P AzC and some mi P Zz0. We have
ρν pgq “

k
ź

pgi smi q νmi .

i“1

Note that
gi smi Fix µ “ gi Fix µ.
So, by Claim 1 and Lemma 3.7 again, the Key Claim follows.
Case (2)-2. L˚ “ A˚Innpsq : CÑC s , and the subgroups C and C s are not conjugate in A.
If g “ ps˚ qm P Q, then
ρν pgq “ pνsqm
Claim 1 (iii) implies that tr2 ˝ρν pgq R T whenever }ν} " 0.
So we may assume g P Q satisfies
k
ź

g“

gi ps˚ qmi

i“1

for some k ě 1, mi P Zzt0u and gi P Vzt1u. Using the normal form theorem for
HNN-extensions, we may assume that g has no subwords of the form s˚ cps˚ q´1 or
ps˚ q´1 c1 s˚ for c P C or c1 P C s . We can write
ρν pgq “

k
ź

gi pνsqmi “

i“1

`
ź

h j νn j

i“1

for some
k, ` ě 1,

gi P Az1,

h j P Lz1,

mi , n j P Zz0.

Suppose Fix µ X h j Fix µ ‰ ∅ for some j. There are five cases.
(I) h j “ s˘1 .
(II) ¨ ¨ ¨ νpsgi qν ¨ ¨ ¨ “ ¨ ¨ ¨ νh j ν ¨ ¨ ¨ .
(III) ¨ ¨ ¨ ν´1 pgi s´1 qν´1 ¨ ¨ ¨ “ ¨ ¨ ¨ ν´1 h j ν´1 ¨ ¨ ¨ .
(IV) ¨ ¨ ¨ ν´1 pgi qν ¨ ¨ ¨ “ ¨ ¨ ¨ ν´1 ph j qν ¨ ¨ ¨ .
(V) ¨ ¨ ¨ νpsgi s´1 qν´1 ¨ ¨ ¨ “ ¨ ¨ ¨ νph j qν´1 ¨ ¨ ¨ .
By Claim 1 (iii), the case (I) is excluded. If (II) occurs, then we have sgi “ h j . If
sgi P Npµq, then
C sgi “ C,

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

47

which contradicts the assumption that C and C s are not conjugate in A. So we have
a nontrivial parabolic element
´1
rc, g´1
i s csgi s

for each c P Cz1. This contradicts the parabolic-free assumption on A, since s´1 cs P
A. Suppose (III) occurs and h j “ gi s´1 . As in the case (II), we have a nontrivial
parabolic element
rc, gi s´1 csg´1
i s P A
for each c P Cz1, which is a contradiction.
Suppose we have (IV). Assuming the minimality of k, we have h j “ gi P AzC;
this is a contradiction to Claim 1 (ii). In (V), we have gi “ h sj P AzC s . If h j P Npµq,
then we would have
C sgi “ C h j s “ C s ,
which contradicts the hypothesis that C s is malnormal in A. So for each c P C, we
have a nontrivial parabolic element
´1 ´1 s
s
´1
s
s ´1
rc, h j ch´1
j s “ rc, sgi s csgi s s “ rc , gi c gi s P A.

This is a contradiction, and the Key Claim is proved.



4.3. Almost Faithful Paths. Let L be a finitely generated group and let ρ0 P Xproj .
We say a path tρt utPR in HompL, PSL2 pRqq is an almost faithful path if ρt P Xproj pLq
for all but countably many t. In this section, we will describe a method to find
an almost faithful path starting from an arbitrary ρ0 P Xproj pLq. In general, it is
not be possible to require that such a path consists of only faithful representations
(Proposition 4.12).
Definition 4.8. Let S be a finite type hyperbolic 2–orbifold. We say a simple closed
curve α is splitting if the Euler characteristic of each component of S zα is negative.
If there exists at least one splitting curve, we say S (and also πorb
1 pS q) is splittable.
For instance, every non-separating essential simple closed curve is splitting.
Lemma 4.9. Let L “ πorb
1 pS q for some finite type hyperbolic 2–orbifold S , and let
α be a splitting curve on S . If ρ0 P Xproj pLq, then there exists an analytic path
tρt : t P Ru Ď HompL, PSL2 pRqq
satisfying the following:
for each b P L, the analytic function tr2 ˝ρt pbq is constant on t if and
only if the free homotopy classes of rαs and b are disjointly realized.
Note we allow the representation ρ0 to be discrete or indiscrete.
Proof of Lemma 4.9. For C “ xrαsy ď L, we have two cases.

48

S. KIM, T. KOBERDA, AND M. MJ

(i) The curve α is separating on S and
L “ xA, By – A ˚C B
for some A, B ď L. In this case, we let V “ A Y B.
(ii) The curve α is non-separating on S and
L “ xA, sy – A˚CÑs´1Cs
for some A, C ď L and s P L. In this case, we let V “ A.
For b P L, we have b P V L if and only if rαs and b can be disjointly realized. Hence
the conclusion follows from Lemma 4.7.

Splittable Fuchsian groups are characterized as follows.
Lemma 4.10. Let S be a finite type hyperbolic 2–orbifold. Then S is splittable if
and only if the signature of S is not one of the following types:
(i) p0; p, q, rq for some 2 ď p, q, r ď 8 satisfying 1{p ` 1{q ` 1{r ă 1;
(ii) p0; 2, 2, 2, pq for some 3 ď p ď 8;
(iii) p0; 2, 2, 2, 2, 2q.
Proof. When an orbifold S of the types (i) through (iii) is cut along an arbitrary
non-null-homotopic simple closed curve, then at least one component has the type
p0; p, 8q or p0; 2, 2, 8q. This implies S is non-splittable.
In order to show the converse, let us assume S is a finite type hyperbolic 2–
orbifold. If the genus of S is zero and S is not one of (i) through (iii), then one can
find an essential separating simple closed curve α such that neither component of
S zα has the type p0; p, 8q or p0; 2, 2, 8q. So S is splittable. If the genus of S is at
least one, we choose a non-separating curve α so that χpS zαq “ χpS q ă 0.

Recall two projective representations ρ, λ of a group L are tracially disjoint if
tr2 ˝ρpLzT L q X tr2 ˝λpLzT L q “ ∅.
We can make an arbitrarily small faithful tracially disjoint deformation of each
faithful parabolic-free representation of a splittable Fuchsian group:
Theorem 4.11. Let L be a splittable Fuchsian group and let ρ0 P Xproj pLq. Then for
each open neighborhood U Ď HompL, PSL2 pRqq of ρ0 there exists a uncountable,
tracially disjoint set Λ satisfying
ρ0 P Λ Ď U X Xproj pLq
such that every pair of representations in Λ are joined by an almost faithful path.
Proof. Let us fix ρ0 and U as in the hypothesis. We let W Ď R be an arbitrary
countable subset. It suffices to find an almost faithful path tρt u Ď U such that
tr2 ˝ρ1 pLzT L q X W “ ∅.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

49

For some compact hyperbolic 2–orbifold S with χpS q ă 0, we can write
L “ πorb
1 pS q.
Let us first consider the case that BS “ ∅. Since S is splittable, we have a finite
collection of infinite cyclic subgroups
C1 , . . . , Cm ď L
whose generators form a filling set of essential simple closed curves on S .
For each i “ 1, . . . , m, we have a decomposition L “ Gi such that one of the
following hold:
(i) Ci is generated by a separating curve and
Gi “ Ai ˚Ci Bi
for some Ai and Bi . We let Vi “ Ai Y Bi for this i.
(ii) Ci is generated by a non-separating curve and
Gi “ Ai ˚Ci ÑtiCi t´1
i

where the amalgamating map is a conjugation by some ti P L. We let Vi “ Ai
for this i.
Since YiCi is filling, we see that
m
č
ViL “ T L .
i“1

By applying Lemmas 4.7 and 4.9 successively, we obtain the desired path.
Now assume BS ‰ ∅. We double S to obtain
S 1 “ S YBS S
1
1
1
and L1 “ πorb
1 pS q. We extend ρ to ρ P Xproj pL q; one can see this by successively
applying Lemma 4.7, as L1 can be obtained from L by an amalgamated free product,
and then by a sequence of HNN-extensions. Now we can apply the closed case to
have a path ρ1t starting from ρ1 . The restriction of ρ1t to S is the desired.


We note that it is not possible to strengthen (i) of Theorem 4.11 so that all ρt ’s
are faithful:
Proposition 4.12. Let L be a finitely generated group and let tρt utPr0,1s be a path
in HompL, PSL2 pRqq such that ρt is faithful for each t. Then tρt utPr0,1s consists of a
single semi-conjugacy class.
Proof. Let g P Lz1. Then rot ˝ρt pgq P Q{Z for all t if and only if g P T L . Since
rot ˝ρt pgq is continuous on t, it follows that rot ˝ρt pgq is constant on t. By Theorem 2.12, we see the semi-conjugacy class of ρt is constant on t.


50

S. KIM, T. KOBERDA, AND M. MJ

We will provide a strengthening of Theorem 4.11 for the rest of this section.
Lemma 4.13. For 2 ď p ă 8 and 3 ď q ă 8, the group Z p ˚ Zq admits at least
one indiscrete faithful parabolic-free projective representation.
Proof. We put
L “ Z p ˚ Zq “ xa, b | a p “ 1 “ bq y,
Y “ tρ P HompL, PSL2 pRqq : rρpabq, Rotp1qs “ 1u.
Then Y is algebraic and ρpabq is elliptic for each ρ P Y. In particular, each faithful
ρ P Y is indiscrete. For all sufficiently large integer r, we have a (non-faithful)
discrete parabolic-free representation ρr : L Ñ PSL2 pRq defined as the composition
L Ñ L{xxpabqr yy Ñ PSL2 pRq.
For each finite set Q Ď Lz1, let us define
YpQq “ tρ P Y : 4 P tr2 ˝ρpQqu.
By Lemma 4.5, we see ρr P YzYpQq. Lemma 4.1 implies that for a very general ρ
in Y, we have ρ P Xproj pLq is indiscrete.

Lemma 4.14. Every finite type splittable Fuchsian group admits at least one indiscrete faithful parabolic-free projective representation.
Remark 4.15. For a surface group, this lemma is immediate from [48] and [34]. We
give an independent proof for general Fuchsian groups.
Proof of Lemma 4.14. Let S be a compact splittable 2–orbifold and L “ πorb
1 pS q.
If BS ‰ ∅, then a repeated application of Lemmas 4.7 and 4.13 implies Xproj pLq
contains an indiscrete representation. So let us assume S is closed and has the
signature
pg; m1 , . . . , mk q.
Case 1. S is splittable by a separating simple closed curve.
This case occurs when g “ 0, in particular. We have a nontrivial splitting
L “ A ˚xcy B
where c is represented by a separating simple closed curve on S . Since A and B are
(once-punctured) hyperbolic 2–orbifold groups, we see that
NA pCq “ C “ NB pCq.
For Q Ď Lz1, let us define algebraic sets
Y “ tρ P HompL, PSL2 pRqq : rρpcq, Rotp1qs “ 1u,
YpQq “ tρ P Y : 4 P tr2 ˝ρpQqu.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

51

Fix a finite set Q Ď Lz1. Each q P Q is either q “ d P Cz1 or
q “ a1 b1 a2 b2 ¨ ¨ ¨ or q “ b1 a1 b2 a2 ¨ ¨ ¨ ,
for some ai P AzC and bi P BzC. Let us denote by C0 , A0 , B0 as the finite sets of
such d’s, ai ’s or bi ’s, respectively. We have the natural quotient maps
α p : A Ñ A{xxc p yy,

β p : B Ñ B{xxc p yy

for each p ą 0. By applying Lemma 4.5 for the elements in A0 , B0 , C0 and also for
the elements rai , cs, rbi , cs, we see that there exists some p " 0 such that
α p pA0 q X α p xcy “ ∅,

1 R α p pC0 q,

β p pB0 q X β p xcy “ ∅,

1 R β p pC0 q.

We fix cocompact Fuchsian representations
A{xxc p yy

ᾱ

/

PSL2 pRq o

β̄

B{xxc p yy,

so that c maps to Rotp2π{pq. We have a commutative diagram as below
/

A{xxc p yy

?A

ᾱ

/
$

L “: A ˚C B

xcy  o


Dρ p

/
5

)

PSL2 pRq

β̄

B

/

p

B{xxc yy

We put
L p “ ρ p pLq “ xρ p pAq, ρ p pBqy.
Note that ρ p pCq is a maximal abelian subgroup of ρ p pAq, as is the case of A{xxc p yy.
By applying Lemma 4.7 to L p , we have a faithful parabolic-free representation
ρ1p : ρ p pAq ˚ρ p pCq ρ p pBq Ñ PSL2 pRq
such that
tr2 ˝ρ1p ˝ ρ p pgq “ tr2 ˝ρ p pgq
for g P A or g P B. Define ρ˚p as the composition
L Ñ ρ p pAq ˚ρ p pCq ρ p pBq Ñ PSL2 pRq.
Then 4 R tr2 ˝ρ˚p pQq and ρ˚p pcq “ Rotp2π{pq P SOp2q, up to conjugacy. So YpQq is
proper in Y. By Lemma 4.1, a very general point ρ P Y belongs to P Xproj pLq and is
indiscrete.
Case 2. S is splittable by a non-separating simple closed curve.

52

S. KIM, T. KOBERDA, AND M. MJ

This case occurs exactly when g ě 1. We proceed in a similar way to the Case 1.
We have a nontrivial splitting
L “ A˚CÑC 1 “ xA, sy
where C “ xcy is represented by a non-separating simple closed curve on S , and
C 1 “ s´1Cs. Note A is a (twice-punctured) hyperbolic 2–orbifold group and
NA pCq “ C.
We define Y and YpQq as in Case 1.
Consider a finite set Q Ď Lz1. Each q P Q can be written as
q “ a1 sm1 a2 sm2 ¨ ¨ ¨ or q “ sm1 a1 sm2 a2 ¨ ¨ ¨ ,
for some ai P Az1 and mi P Zz0. By Lemma 4.5, if p is sufficiently large then the
natural quotient map
α p : A Ñ A{xxc p , pc1 q p yy
sends each ai to a nontrivial element. We can further require that whenever ai R xcy,
we have α p pai q R xα p pcqy, and whenever ai R xc1 y, we have α p pai q R xα p pc1 qy,
We fix cocompact Fuchsian representations
ᾱ : A{xxc p , pc1 q p yy Ñ PSL2 pRq
so that c and c1 map to rotations of angle 2π{p; in particular, we have some s̄ P
PSL2 pRq such that
ᾱpc1 q “ s̄´1 ᾱ p pcq s̄.
We have a commutative diagram as below
:

/

A

A{xxc p , pc1 q p yy
ᾱ

'

C ˚ C1

L“
A˚CÑC 1
8
$

Dρ p

/

)
5

PSL2 pRq

sÞÑ s̄

C ˚ xsy

/

p

C{xxc yy ˚ xsy

We put
L p “ ρ p pLq “ xρ p pAq, ρ p psqy.
Since ρ p pCq is a maximal abelian subgroup of L p , we can apply Lemma 4.7 to
obtain a faithful parabolic-free representation
ρ1p : ρ p pAq˚ρ p pCqÑρ p pC 1 q Ñ PSL2 pRq
such that
tr2 ˝ρ1p ˝ ρ p pgq “ tr2 ˝ρ p pgq

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

53

for g P A. Define ρ˚p as the composition
L Ñ ρ p pAq˚ρ p pCqÑρ p pC 1 q Ñ PSL2 pRq.
Then 4 R tr2 ˝ρ˚p pQq and ρ˚p pcq “ Rotp2π{pq P SOp2q. So YpQq ‰ Y. By
Lemma 4.1, a very general point ρ P Y is indiscrete and belongs to Xproj pLq.

Let S g denote a closed oriented hyperbolic surface of genus g. It follows from [74]
that each discrete faithful representation
ρ0 : π1 pS g q Ñ PSL2 pRq
is locally rigid in Hompπ1 pS g q, Homeo` pS 1 qq. This means that an open neighborhood U of ρ0 is a single semi-conjugacy class. Here we prove that each indiscrete
ρ0 P Xproj pLq of a splittable Fuchsian group L is locally flexible in the following
strong sense:
Theorem 4.16. Let L be a splittable Fuchsian group.
(1) Then there exists uncountably many pairwise-inequivalent indiscrete faithful parabolic-free projective representations of L.
(2) Let ρ be an indiscrete faithful parabolic-free projective representation of
L. Then every open neighborhood U Ď HompL, PSL2 pRqq of ρ contains
a uncountable set Λ Ď U X Xproj pLq of pairwise-inequivalent indiscrete
representations such that each pair of points in Λ can be joined by an almost
faithful path in U.
Proof. By Lemma 4.14, there exists at least one ρ as in part (2). The conclusion
follows from Theorem 4.11.

Remark 4.17. For a surface group L, there exists a dense set of faithful representations in
HompL, PSL2 pKqq,
as shown in [34]. Let ρ0 : L Ñ PSL2 pKq be a representation. A small modification
of a Baire category argument in the same paper implies that ρ0 admits an arbitrarily
small deformation tρt utPr0,1s such that the property (ii) of Theorem 4.11 holds, and
from this, one can then deduce each indiscrete representation in Xproj pLq in Theorem 4.16 is accumulated. However, such an approach does not directly yield almost
faithful paths or work with the orbifold case.
Remark 4.18. Let L “ π1 pS g q. For each ρ P HompL, Homeo` pS 1 qq, the Milnor–
Wood Inequality asserts that eupρq takes integer values between 2 ´ 2g and 2g ´ 2.
Moreover, |eupρq| “ 2g ´ 2 if and only if ρ is injective and semi-conjugate to a
discrete faithful representation of L into PSL2 pRq [48, 78, 74]. On the other hand,
a connected component of
Hompπ1 pS g q, PSL2 pRqq

54

S. KIM, T. KOBERDA, AND M. MJ

is precisely determined by its Euler number [48]. Each component contains a faithful parabolic-free (possibly indiscrete) representation by [34]. So Theorem 4.16
implies that each component where the Euler number is not ˘p2 ´ 2gq contains uncountably many distinct semi-conjugacy classes of faithful parabolic-free actions
which are joined by almost faithful paths.
Example 4.19. Theorem 4.16 is not true without the splittability hypothesis. For
example, consider a hyperbolic triangle group
L “ xa, b, c | a p “ bq “ cr “ abc “ 1y
such that 2 ď p, q, r ă 8 and 1{p ` 1{q ` 1{r ă 1. If ρ : L Ñ PSL2 pRq is a
representation, then the elliptic centers of ρpaq, ρpbq, ρpcq form a hyperbolic triangle
with internal angles
uπ{p, vπ{q, wπ{r,
for some u, v, w P N. Since pu, v, wq determines the isometry type of the triangle formed by the centers of those rotations, there are only finitely many possible
conjugacy class of a representations ρ : L Ñ PSL2 pRq. By a similar (and easier)
reasoning, we see that an infinite dihedral group does not admit an indiscrete representations.
4.4. Simultaneous Control of Rotation Numbers. So far we have developed techniques of finding uncountably many inequivalent faithful circle actions of a given
group. In this subsection, we describe an approach to simultaneously control the
rotation numbers of given (finitely many) group elements.
To state the main result of this section more precisely, let us introduce a notation.
Suppose L is a finitely generated group and T is a set of elements (or, conjugacy
classes) in L. We write
EpL, T q “ tρ P Xproj pLq : rot ˝ρpT q is a singletonu.
A multi-curve T on a 2–orbifold S means a finite collection of disjoint simple
closed curves on S , which can be identified with a certain set of conjugacy classes
orb
in πorb
1 pS q. For each representation ρ : π1 pS q Ñ PSL2 pRq we define
rot ˝ρpT q “ trot ˝ρpgq : rgs P T u Ď R{Z.
Theorem 4.20. Let S be a finite type hyperbolic 2–orbifold and let T be a multicurve on S . If each element of T is either splitting or peripheral, then the following
set is uncountable:
trot ˝ρpT q : ρ P Epπorb
1 pS q, T qu.
Hence, there exist uncountable set Λ of pairwise-inequivalent indiscrete faithful
parabolic-free projective representations of πorb
1 pS q such that each ρ P Λ maps T to

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

55

elliptic elements of the same nonzero rotation number. So we have the following
corollary. Recall
MpLq “ OutpLqz HompL, Homeo` pS 1 qq{semi-conjugacy.
Corollary 4.21. If S is a finite type hyperbolic surface and if T is a multi-curve on
S , then the image of Epπ1 pS q, T qq in Mpπ1 pS qq is uncountable.
We will prove Theorem 4.20 in this subsection. Recall our notation
xxT p yy “ xxtg p : rgs P T uyy E πorb
1 pS q.
Lemma 4.22. Let S and T be as in Theorem 4.20.
p
(1) If rcs P T is peripheral on S , then the group πorb
1 pS q{xxT yy contains xcy –
Z p as a maximal abelian subgroup.
(2) For distinct peripheral c1 , c2 P T , the images of xc1 y and xc2 y are nonp
conjugate in πorb
1 pS q{xxT yy.
(3) For all sufficiently large p ą 0, there exists a faithful parabolic-free reprep
sentation φ p : πorb
1 pS q{xxT yy Ñ PSL2 pRq such that
rot ˝φ p pT q “ t1{pu.
p
Proof. Put L “ πorb
1 pS q, and define L p “ L{xxT yy.
The parts (1) and (2) follow from the normal form theorem for graph of groups [69].
Let us consider part (3). If all elements in T are peripheral, then we can find a
parabolic-free Fuchsian representation φ p such that the curves in T are mapped to
rotations of angle 2π{p, while the other peripheral elements are mapped to hyperbolic elements. So we may assume some element c P T is non-peripheral. We will
induct on the complexity ξpS q ą 0.
Suppose c is separating. We have a nontrivial decomposition L “ A ˚C B for
C “ xcy. Possibly after conjugation, we may write

T A Y T B “ T,

T A X T B “ tcu.

By the inductive hypothesis on each component of S zc, we have faithful parabolicfree representations
A p “ A{xxT Ap yy

αp

/

PSL2 pRq o

βp

B p “ B{xxT Bp yy.

such that
α p pcq “ β p pcq “ Rotp2π{pq.
By part (1), we can apply the Pulling-Apart Lemma to the group
xA p , B p y
to have the desired representation L p Ñ PSL2 pRq; see Lemma 4.7 (i).

56

S. KIM, T. KOBERDA, AND M. MJ

If c is non-separating, the proof is almost identical using part (2) and the PullingApart Lemma (iii).

The following composition yields a stably injective sequence of parabolic-free
representations:
φ
/ πorb pS q{xxT p yy p

πorb
1 pS q

1

/

PSL2 pRq

Lemma 4.23. Let L be a finitely generated group and let T Ď L be a nonempty
finite set. Suppose that for all sufficiently large p there exists c p P p0, 1{ps and a
parabolic-free projective representations ρ p such that
rot ˝ρ p pT q “ tc p u
and such that the sequence tρ p u is stably injective. Then the following set is uncountable:
trot ˝ρpT q : ρ P EpL, T qu.
Proof. Let us fix a finite set A Ď LzT so that L “ xT Y Ay. Define an algebraic set
Y ˚ “ SOp2q ˆ PSL2 pRqT ˆ PSL2 pRqA .
Consider an arbitrary
χ “ pα0 , tαt utPT , tβa uaPA q P Y ˚ .
We will denote the projection map to the first component
χ ÞÑ α0
as p0 . There exists at most one homomorphism
φχ P HompL, PSL2 pRqq
satisfying the following:
#
αα0 x ,
φχ pxq “
βx ,

if x P T
if x P A.

So we have an algebraic set
Y “ tχ P Y ˚ : φχ defines a homomorphismu,
We can find χ p P Y such that
ρ p “ φχ p ,

rot ˝ρ p pT q “ p0 pχ p q.

For each finite subset Q Ď Lz1, we have an algebraic set
YpQq “ tχ P Y : φχ pQq contains a (possibly trivial) parabolic elementu.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

57

We may identify
ď

EpL, T q “ Yz

YpQq.

QĎLz1, finite

Let W Ď SOp2q be an arbitrary countable set. For each finite subset U Ď W, we
define
WpUq “ p´1
0 pUq X Y.
Since tρ p u is stably injective and parabolic-free, each YpQq is proper in Y. Moreover, the condition on the rotation number implies that WpUq is proper as well. By
Lemma 4.1, the following set is uncountable:
ď
WpUq.
EpL, T qz
UĎW, finite



It follows that p0 pEpL, T qq is uncountable.
Remark 4.24. In Theorem 4.23, we have the condition that
rot ˝ρ p pT q “ tc p u.

More generally, one can modify this condition to the following:
For some k ą 0 and a kˆT integer matrix B, we have B¨rot ˝ρ p pT q “
0 P Rk .
Here we regard rot ˝ρ p pT q as an |T |–dimensional vector. For a proof, we have only
to modify Y ˚ as
Y ˚ “ tpα, βq : α P V and β P PSL2 pRqB u.
for a suitable algebraic set
V Ď pSOp2q ˆ PSL2 pRqqT .
See Lemma 6.14 for a generalization along this line.
Example 4.25. Let L be the fundamental group of a genus–two surface. For an
illustrative purpose, let us prove that the image of Xproj pLq is uncountable in MpLq,
without resorting to Theorem 4.16. See Lemma 6.14 for a generalization.
We have a sequence of maps as follows.
L “ xa, b, c, d|ra, bs “ rc, dsy 



ι

/

H “ xa, b, t|rra, bs, ts “ 1y
fp

ιpcq “ at , ιpdq “ bt

gq

f p ptq “ ra, bs p



F2 “ xa, by



q
πorb
1 pT q q “ xa, b|ra, bs “ 1y

 ρq
/

PSL2 pRq.

58

S. KIM, T. KOBERDA, AND M. MJ

Here, gq is the natural quotient map, and T q is a compact genus–one hyperbolic
orbifold with exactly one cone point, whose order is q. The group πorb
1 pT q q admits
a cocompact Fuchsian representation ρq such that
ρq ra, bs “ Rotp2π{qq.
Since f p and gq are stably injective by Lemmas 3.1 and 4.5, for each p ě 1 we can
choose a sufficiently large
q “ qppq ě p
such that
(
φ p “ gqppq ˝ f p : H Ñ πorb
1 pT qppq q
is a stably injective sequence. Note that
rot ˝φ p ra, bs “ 1{q.
So we can apply Theorem 4.23 and see that the following set is uncountable:
trot ˝ρ pra, bsq : ρ P EpH, tra, bsuqu .
In particular, L admits uncountably many pairwise-inequivalent indiscrete faithful
parabolic-free representations.
5. Combination Theorem for Flexible Groups
In this section, we establish a combination theorem (Theorem 5.5) for the class
of flexible and liftable-flexible groups.
5.1. Statement of the result. Recall that a set
Λ Ď HompL, PSL2 pRqq
is tracially disjoint if for all distinct ρ and λ in Λ we have
tr2 ˝ρpLzT L q X tr2 ˝λpLzT L q “ ∅.
The following definition is given in the introduction.
Definition 5.1. A finitely generated group L is called flexible if there exists a uncountable, tracially disjoint subset Λ of indiscrete faithful parabolic-free projective
representations. If, furthermore, Λ can be chosen so that each representation in Λ
is liftable, then we say L is liftable-flexible.
We will often use the following alternative characterization of flexibility. The
proof is immediate by Zorn’s lemma.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

59

Lemma 5.2. A finitely generated group L is flexible if and only if for each countable
set W Ď R there exists an indiscrete faithful representation ρ : L Ñ PSL2 pRq such
that
tr2 ˝ρpLzT L q X W “ ∅.
Moreover, L is liftable-flexible if and only if such ρ can be chosen to be liftable.
Example 5.3.
(1) The infinite cyclic group is liftable-flexible. This is because
we can map a generator to an arbitrary irrational rotation.
(2) Infinite dihedral groups and triangle groups are not flexible (Example 4.19).
Let us note several restrictions for flexible groups. The proofs are trivial.
Proposition 5.4.
(1) A flexible group is commutative–transitive.
(2) Every finite subgroup of a flexible group is cyclic.
(3) A liftable-flexible group is torsion-free.
We will denote by F and F „ the classes of flexible groups and liftable-flexible
groups, respectively. Note that F „ Ď F . The main result of this section is the
following.
Theorem 5.5 (Combination Theorem for Flexible Groups). Let H “ F or F „ .
(1) Infinite cyclic groups are in H .
(2) If H ď L P H and rL : Hs ă 8, then H P H .
(3) If A and B are in H , then so is A ˚ B.
(4) Let A, B ď L and C ď A X B satisfy that
L “ xA, By P H .
If C is nontrivial malnormal abelian in A and in B, then A ˚C B P H .
(5) Let A, C ď L and s P L satisfy that
L “ xA, sy P H
and that C and C s are nontrivial malnormal abelian subgroups of A. Assume either
‚ s P ZpCq, or
‚ C and C s are not conjugate in A.
Then A˚Innpsq : CÑC s P H .
(6) If A is in F and B is a finite cyclic group, then A ˚ B P F .
Remark 5.6. The malnormality condition of C can be replaced by a seemingly
weaker (but equivalent in our case) condition that
NL pCq X V “ C,
where V “ A Y B in (1) and V “ A in (2).

60

S. KIM, T. KOBERDA, AND M. MJ

5.2. Proof.
Proof of Theorem 5.5. Throughout this proof, we will fix a countable set W Ď R.
Our goal in each part is, to find ρ satisfying the conditions in Lemma 5.2. We will
first consider the case H “ F .
Part (1) is shown in Example 5.3. For part (2), suppose ρ0 : L Ñ PSL2 pRq is
indiscrete faithful and satisfies
ρ0 pLzT L q X W “ ∅.
By Lemma 2.24, we can pick g0 P L such that rot ˝ρpg0 q is irrational. There is
N ą 0 such that
h0 “ g0N P H.
So λæH is the desired indiscrete faithful representation.
In the Pulling-Apart Lemmas, we note that if g is an irrational rotation in L ď
PSL2 pRq, and if ν is sufficiently close to the identity, then very generally ρν pgq is
also an irrational rotation. This is because ρν is continuous on ν and g has infinite
order as an element of L. So the parts (3) through (5) are immediate from the
Pulling-Apart Lemma (Lemma 4.7).
Part (6) follows from Lemma 4.6 (1).
Now, the case H “ F „ follows from the lemma below.

Lemma 5.7. Let G, G̃ be groups and p : G̃ Ñ G be an epimorphism. Choose an
abelian subgroup µ ď G such that µ̃ “ p´1 pµq is abelian, and fix
ν P µ,

ν̃ P µ̃.

For some group L, suppose we have maps
ρ : L Ñ G,

ρ̃ : L Ñ G̃,

such that p ˝ ρ̃ “ ρ.
(1) Let A, B ď L and C ď A X B satisfy L “ xA, By and ρpCq ď µ. If we put
L˚ “ A ˚C B,
then there uniquely exist homomorphisms
ρ˚ : L˚ Ñ G,

ρ̃˚ : L˚ Ñ G̃

such that the following hold:
(i) ρ˚ æA “ ρæA and ρ˚ æB “ Innpνq ˝ ρæB ;
(ii) ρ̃˚ æA “ ρ̃æA and ρ̃˚ æB “ Innpν̃q ˝ ρ̃æB ;
(iii) p ˝ ρ̃˚ “ ρ˚ .

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

61

(2) Let A, C ď L and s P L satisfy C Y C s ď A, L “ xA, sy and ρpCq ď µ. If we
put
L˚ “ A˚CÑs´1Cs ,
then there uniquely exist homomorphisms
ρ˚ : L˚ Ñ G,

ρ̃˚ : L˚ Ñ G̃

such that the following hold:
(i) ρ˚ æA “ ρæA and ρ˚ psq “ νρpsq;
(ii) ρ̃˚ æA “ ρ̃æA and ρ̃˚ psq “ ν̃ρ̃psq;
(iii) p ˝ ρ̃˚ “ ρ˚ .
Proof. The proof is immediate from the universality in the following diagrams.
?A

ρ
ρ̃



C

?L

ρ̃˚

˚

)/
5

Innpν̃q˝ρ̃



2/,

p

G̃

G

Innpνq˝ρ

B
9

A
ρ

#

C ˚ Cs

ρ̃

;L

˚

ρ̃˚

*/
4

G̃

,2/

p

sÞÑν̃ρ̃psq

G

sÞÑνρpsq

$

C ˚ xsy

For the rest of this section, we will investigate consequences of the combination
theorem and the properties of (liftable-)flexibility.
A particularly simple cases of the combination theorem are as below. Let A be a
group and C be a subgroup. Then we simply write
A˚C “ A˚Id : CÑC “ A ˚C pC ˆ Zq.
For a group A and a subgroup C, we denote by A ˚C Ā the amalgamation of two
copies of A along IdC .
Corollary 5.8. Let A be a group and let C be an abelian and malnormal subgroup
of A. If A is flexible or liftable-flexible, then so are A˚C and A ˚C Ā.
Proof. Apply Combination Theorem (3), (4), (5) for L “ A “ B and s “ 1.
Let us observe a consequence for Fuchsian groups.



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S. KIM, T. KOBERDA, AND M. MJ

Corollary 5.9.
(1) Finite type splittable Fuchsian groups are flexible.
(2) Finite type torsion-free Fuchsian groups are liftable-flexible.
Proof. Part (1) follows from Theorem 4.11 and Theorem 4.16. For part (2), let L be
a torsion-free Fuchsian group of finite type. If L is free, then Theorem 5.5 (1) and
(3) imply that L is liftable-flexible. If L is not free, we may assume
L “ π1 pS 2 q “ F2 ˚Z F2 “ xa, by ˚ra,bs“rc,ds xc, dy,
using Theorem 5.5 (2). In this case, Corollary 5.8 implies L is liftable-flexible.



5.3. Exotic Circle Actions. In this section we provide uncountably many distinct
exotic circle actions of flexible groups, and make a connection to the existence of
quasi-morphisms on those groups. We will prove that most Fuchsian groups and
their iterated generalized doubles are flexible. Limit groups will be shown to be
liftable-flexible.
Recall there is an embedding
HompL, Homeo` pS 1 qq{semi-conjugacy Ñ Hb2 pL; Zq
defined by ρ ÞÑ ρ˚ eub .
Lemma 5.10. Let L be a countable group and g0 P L. Suppose
Λ Ď HompL, PSL2 pRqq
satisfies that
trot ˝ρpg0 q : ρ P Λu
is Z–linearly independent. Then
tρ˚ eub : ρ P Λu
is linearly independent in Hb2 pL; Zq.
Proof. Let us suppose that
k0 ρ˚0 eub

n
ÿ

“

ki ρ˚i eub

i“1

for some ρi P Λ and nonzero integers ki . Then there exists a bounded map β : G Ñ
Z such that
n
ÿ
˚
k0 ρ0 eu “
ki ρ˚i eu ` Bβ
i“1

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

63

as two–cocycles. By Lemma 2.1, we have
N
1 ÿ ˚
ρ0 eupg0 , g0j´1 q
NÑ8 N
j“1
˜
¸
N
n
ÿ
ÿ
1
“ lim
ki ρ˚i eupg0 , g0j´1 q ` βpg0 q ` βpg0j´1 q ´ βpg0j q
NÑ8 k0 N
j“1 i“1

rot ρ0 pg0 q “ lim

n
ÿ
ki
1
rot ρi pg0 q.
“ βpg0 q `
k0
k
i“1 0

This contradicts the Z-linear independence hypothesis.



Corollary 5.11. If L is a flexible group, then there exists a uncountable set Λ of
faithful parabolic-free projective representations of L such that the following hold.
(i) All representations in Λ stay distinct in the double coset space
MpLq “ OutpLqz HompL, Homeo` pS 1 qq{semi-conjugacy.
(ii) The image of Λ is Z–linearly independent in Hb2 pL; Zq.
(iii) If, furthermore, L is liftable-flexible, then Λ maps to the kernel of
Hb2 pL, Zq Ñ H 2 pL, Zq.
Proof. Let Λ0 be a maximal tracially disjoint set of indiscrete representations in
Xproj Ď HompL, PSL2 pRqq. By flexibility, the set Λ0 is uncountable and L ‰ T L .
For two distinct ρ, λ P Λ0 , the two sets
rot ˝ρpLqzpQ{Zq,

rot ˝λpLqzpQ{Zq

are a disjoint pair of infinite subsets of R{Z. Hence Λ0 satisfies the condition (i).
For each ρ P Λ0 , there exists some g P L such that ρpgq is an elliptic element of
infinite order. Since Λ0 is uncountable and L is countable, we have some g0 P L
such that the following set is uncountable:
Λ1 “ tρ P Λ0 : rot ˝ρpg0 q R Q{Zu.
By our assumption, for all distinct ρ, λ P Λ1 , we have
rot ˝ρpg0 q ‰ rot ˝λpg0 q.
Since the Z–span of a countable set in R{Z is countable, we may further assume
that the following set is Z–linearly independent:
trot ˝ρpg0 q : ρ P Λ1 u.
By Lemma 5.10, we see that Λ1 satisfies (ii).

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S. KIM, T. KOBERDA, AND M. MJ

If L is liftable-flexible as in (iii), we will further require that each representation
in Λ0 is liftable. In this case, Λ0 maps to the kernel of
Hb2 pL, Zq Ñ H 2 pL, Zq.

5.4. Quasi-Morphisms. It is known that HQMpL; Zq is infinite dimensional if L
is a nontrivial free group, a non-elementary word-hyperbolic group or a mapping
class group [18, 37, 7]. Our construction shows that for a liftable-flexible L, the
abelian group HQMpL; Zq has uncountable dimension even when we count only
defect–one subadditive quasimorphisms:
Corollary 5.12. If L is liftable-flexible, then there exists a set Λ of integer–valued
subadditive defect–one quasimorphisms such that
trλs P HQMpL; Zq : λ P Λu
is uncountable and linearly independent.
Proof. This is immediate from Corollary 5.11 (ii), along with Lemmas 2.19 and 5.10.

5.5. Limit Groups.
Definition 5.13 (cf. [30]).
(1) An epimorphism φ : L Ñ L1 is called a generalized double over L1 if L “ A ˚C B for some finitely generated nontrivial
groups A and B such that C is maximal abelian both in A and in B, and such
that φæA and φæB are both injective.
(2) An epimorphism φ : L Ñ L1 is called a centralizer extension if L “ A˚C for
some finitely generated nontrivial group A such that C is maximal abelian
in A and such that φæA is injective.
(3) If there is a sequence of epimorphisms
L “ Ln

fn

//

Ln´1

fn´1

//

¨¨¨

//

L1

f1

//

L0 ,

such that each fi is a generalized double or a centralizer extension, then we
say L is an iterated generalized double over L0 and write L P IGDpL0 q.
(4) In part (3), we say a group L is an iterated malnormal generalized double
over L0 if the amalgamating subgroup of Li in each step is further required
to be malnormal in the vertex groups of Li . We write L P IMGDpL0 q in this
case.
The following characterizes limit groups as iterated generalized doubles of free
groups.
Lemma 5.14.

(1) Every limit group is in IMGDpFr q for some r.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

65

(2) Suppose L and H are groups such that L is in IMGDpHq. If H is flexible or
liftable-flexible, then so is L.
Proof. (1) It is well-known that every limit group L is torsion-free and in IGDpFr q
for some r; see [93, 30]. Moreover, L embeds into PSL2 pRq [8, 102]. For a torsionfree group L ď PSL2 pRq, a subgroup C is maximal abelian if and only if C is
malnormal (Remark 1.5). Hence the sequence in Definition 5.13 (3) ending with Fr
is actually an iterated malnormal generalized double.
Part (2) is immediate from the Combination Theorem.

Recall Fr is liftable-flexible (Theorem 5.5). By Lemmas 4.7 and 5.14, we have
the following general result.
Corollary 5.15.
(1) Every nontrivial limit group is liftable-flexible.
(2) Every nontrivial limit group admits a set Λ of integer–valued subadditive
defect–one quasimorphisms such that
trλs P HQMpL; Zq : λ P Λu
is uncountable and linearly independent.
Consider the special case of part (1) in Corollary 5.15, when L is non-Fuchsian.
Then we can give a shorter proof of the flexibility of L as follows; see [102, 5] for
similar ideas. Let Q Ď Lz1 and W Ď R be finite sets such that 4 P W. We can find
homomorphisms
L

λ

/

F2

ρ

/

PSL2 pRq

such that 1 R λpQq and such that tr2 ˝ρ ˝ λpQq X W “ ∅, using Projective Baumslag
Lemma. A Baire Category argument as in Lemma 4.3 shows that for all countable
W Ď R there exists a faithful parabolic-free projective representation ρ of L such
that
tr2 ˝ρpLz1q X W “ ∅.
Since L is non-Fuchsian, we see that ρ is necessarily indiscrete, and so, L is flexibile. However, this argument neither yields directly the liftable–flexibility of L, nor
applies to the case when L is Fuchsian.
6. Axiomatics
6.1. Tracial Structure. A map defined on a group is called a class function if the
map is constant on each conjugacy class.
Definition 6.1. Let G be a commutative–transitive topological group, which is
either a complete metric space or a locally compact Hausdorff space. Suppose
ϕ : G Ñ S is a continuous surjective class function for some Hausdorff space

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S. KIM, T. KOBERDA, AND M. MJ

S . Under this setting, we say the pair pG, ϕq is a tracial structure if both of the
following hold.
(A) Baumslag I. Let f : G Ñ G be a map defined by
(i) f pνq “ νm for some m P Zz0, or
(ii) f pνq “ g1 νm1 g2 νm2 ¨ ¨ ¨ gk νmk for some k ě 1, mi P Zz0 and gi P Gz1.
Then for each x P S , the set pϕ ˝ f q´1 pxq is nowhere dense in G.
(B) Baumslag II. Let µ be a maximal abelian subgroup of G such that ϕpµq ‰
tϕp1qu. Suppose a map f : µ Ñ G is defined by
(i) f pνq “ νm for some m P Zz0, or
(ii) f pνq “ g1 νm1 g2 νm2 ¨ ¨ ¨ gk νmk for some k ě 1, mi P Zz0 and gi P G such
that for each i, there exists c P µ satisfying ϕrc, gi cgi ´1 s ‰ ϕp1q.
Then for each x P S , the set pϕ ˝ f q´1 pxq X µ is nowhere dense in µ.
Example 6.2. The reader is advised to keep in mind the key examples of this paper,
which are G “ PSL2 pKq for K “ R or K “ C. We let
ϕ “ tr2 : PSL2 pKq Ñ K,
The conditions Baumslag I, II follow from Baumslag Lemma for PSL2 pKq, that is
Lemma 3.7. It is routine to verify the rest of the conditions.
pkq

Example 6.3. The group PSL2 pRq for 2 ď k ď 8 admits a natural tracial structure:
pkq

tr2 : PSL2 pRq Ñ PSL2 pRq Ñ R.
Definition 6.4. Given a tracial structure pG, ϕ : G Ñ S q, a finitely generated group
L is in the class F pG, ϕq if for each nonempty open set U Ď S there exists a
uncountable subset Λ Ď HompL, Gq of faithful representations such that ρpLqXU ‰
∅ for each ρ P Λ and such that
ϕ ˝ ρpLzT L q X ϕ ˝ λpLzT L q “ ∅
for all distinct ρ and λ in Λ.
Example 6.5. A flexible group (Definition 5.1 and Lemma 5.2) is in the class
F pPSL2 pRq, tr2 q. To see this, recall that a flexible group L admits an indiscrete
faithful representation. If L is virtually cyclic, then L is actually cyclic (Example 4.19) and its generator can assume a very general trace value in R. If L contains
F2 , then the the trace spectrum of an arbitrary faithful indiscrete representation is
dense in R (Lemma 2.24).
For the rest of this section, we fix a tracial structure pG, ϕ : G Ñ S q, and simply
write Fϕ “ F pG, ϕq. We set P “ ϕ´1 ˝ ϕp1q and call each element in P as a parabolic. A set is parabolic-free if it does not contain a nontrivial parabolic element.
A subgroup H ď G will be called non-parabolic if H ­Ď P. The Combination
Theorem for Flexible Groups (Theorem 5.5) generalizes as follows.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

67

Theorem 6.6.
(1) Infinite cyclic groups are in Fϕ .
(2) If A, B P Fϕ , then A ˚ B P Fϕ .
(3) Let A, B ď L and C ď A X B satisfy that
L “ xA, By P Fϕ .
If C is nontrivial malnormal abelian in A and in B, then A ˚C B P Fϕ .
(4) Let A, C ď L and s P L satisfy that
L “ xA, sy P Fϕ
s

and that C and C are nontrivial malnormal abelian subgroups of A. Assume either
‚ s P ZpCq, or
‚ C and C s are not conjugate in A.
Then A˚Innpsq : CÑC s P Fϕ .
The proof of Theorem 6.6 is almost identical to that of Theorem 5.5, so will be
only sketched. We first point out two key lemmas below needed for the proof.
Lemma 6.7.
(1) A very general ν P G is non-parabolic and has infinite order.
(2) If µ is a maximal abelian subgroup of G, then µ and Npµq are closed and
nowhere dense in G; in particular, they are Baire spaces.
Proof. (1) Apply Baumslag I for f pνq “ νm for m P Zz0.
(2) Fix c P µz1. As was seen in Lemma 2.22, the subgroups µ and Npµq are
algebraically defined as
µ “ tg P G : rc, gs “ 1u,

Npµq “ tg P G : rc, gcg´1 s “ 1u.

By Baumslag II, both are closed and nowhere dense in G.



As before, the process of pulling-apart subgroups will play a crucial role for us.
Lemma 6.8 (General Free Product Pulling-Apart Lemma).
(1) Let A and B be
countable subgroups of G, and let W Ď S be a countable set. Then for
a very general ν P G, the group Lν “ xA, Bν y is isomorphic to A ˚ B and
furthermore,
`
˘
ϕ Lν zpA Y Bν qLν X W “ ∅.
(2) Let A be a countable subgroup of G, and let W Ď S be a countable set.
Then for a very general ν P G, the group Lν “ xA, νy is isomorphic to A ˚ Z
and furthermore,
`
˘
ϕ Lν zALν X W “ ∅.
Proof. Let V “ A Y B in (1), and V “ A in (2). We choose µ to be a non-parabolic
maximal abelian group such that rc, gcg´1 s is not parabolic for all c P µz1 and
g P xVyz1. here, the hypothesis Baumslag I is used. The rest of the proof is very
similar to the proof of Lemma 4.6.


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S. KIM, T. KOBERDA, AND M. MJ

Recall that for C ď G we write
ZpCq “ tν P G : rc, νs “ 1 for all c P Cu.
Lemma 6.9 (General Amalgamated Pulling-Apart Lemma). For a countable subgroup L ď G and for a countable set W Ď S , assume one of the following.
(1) For some subgroups A, B, C ď L, we have that L “ xA, By and that C
is a malnormal nontrivial abelian subgroup of A and also of B. For each
ν P ZpCq we let ρν be the map uniquely determined by the following commutative diagram
?A

IdA

/

%

Co

ρν

A ˚C B
L˚ “
9


+3/

G

Innpνq˝IdB

B

We further assume V “ A Y B satisfies ϕp1q R ϕpVz1q.
(2) For some subgroups A, C ď L and an element s P L, we have that L “
xA, sy and that C and C s are malnormal nontrivial abelian subgroups of A.
Furthermore, we assume that either
‚ s P ZpCq, or
‚ C and C s are not conjugate in A.
For each ν P ZpCq we let ρν be the map uniquely determined by the following commutative diagram
9

A
IdA

+

(

C ˚ Cs

L˚ “ A˚Innpsq : CÑC s
6

%

ρν

/2,

G

s˚ ÞÑνs
˚

C ˚ xs y
Here s˚ denotes the stable generator of L˚ . We further assume V “ A
satisfies ϕp1q R ϕpVz1q.
Under the assumption (1) or (2), for a very general ν P ZpCq the map ρν is faithful
and
´
¯
˚
L˚
ϕ ˝ ρν L zV
X W “ ∅.
The proof of the above lemma is omitted, as it is identical to that of Lemma 4.7,
after substituting the phrase “closed nowhere dense” for “finite”.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

69

Proof of Theorem 6.6. For the proof, let us fix U Ď G, W Ď S such that U is
open and W is countable. Furthermore, we assume ϕp1q P W. We use an alternate
characterization of flexibility as in Lemma 5.2.
(1) In Lemma 6.9 (2), we simply put A “ L “ t1u and ν P ϕ´1 pUq. In
Lemma 6.7, we saw a very general ν has infinite order.
(2) Suppose α : A Ñ G and β : B Ñ G be faithful representations satisfying
the conditions (of ρ) in Definition 6.4. Then Lemma 6.8 (1) applies with L “
xαpAq, βpBqy.
(3) Suppose we have a faithful representation ρ : L Ñ G such that
ρpLq X U ‰ ∅,

ρpLzT L q X W “ ∅.

Fix g P L such that ρpgq P U. Let µ ď G be the maximal abelian subgroup
containing ρpCq. For each ν P µ, we have a map
ρν : L˚ “ A ˚C B Ñ G
as in the proof of the Pulling Apart Lemma (Lemma 4.7). Then for a very general
ν P µ, the representation ρν satisfies
ρν pLzT L q X W “ ∅.
In particular, one can find ν P µ such that ρν pgq P U.
We omit the proof of (4), as it is very similar to the above.



By Lemma 5.14 and Theorem 6.6, we have the following.
Corollary 6.10. Every nontrivial limit group is in Fφ .
6.2. UV-structure. We can also generalize Theorem 4.23 to the context of an arbitrary algebraic group in order to control certain “spectra” of representations. Suppose we want to find many representations ρ : G Ñ PSL2 pRq which map certain
fixed elements to irrational rotations. Note that neither the set E “ ptr2 q´1 p0, 4q Ď
PSL2 pRq nor the map
rot : PSL2 pRq Ñ R{Z
is algebraic. So we use a parametrization by an algebraic group:
ξ

SOp2q ˆ PSL2 pRq

//

E

η

'




/

PSL2 pRq

η̄

SOp2q
where
ξpx, yq “ xy ,

ηpx, yq “ x,

η̄pxq “ Rotp2π rotpxqq.

70

S. KIM, T. KOBERDA, AND M. MJ

Definition 6.11.
(1) Let G be an algebraic group and k ą 0. For each i “
1, 2, . . . , k, we assume there is a commutative diagram
Ui

ξi

/ / ξi pU i q 
ηi

" 


/

G,

η̄i

Vi

such that
(i) Ui and Vi are algebraic sets;
(ii) ξi and
Ťk ηi are polynomial maps;
(iii) Gz i“1 ξi pUi q is an algebraic set.
In this situation, we say G is equipped with a UV–structure.
(2) Assume (1). Suppose G is a finitely generated group, and
A “ tA1 , . . . , Ak u
is a collection of finite subsets Ai Ď G. Fix algebraic subset
ź
Wi Ď ViAi “
Vi
Ai

for each i “ 1, 2, . . . , k, and put
W “ tW1 , . . . , Wk u.
We say a representation ρ : G Ñ G is UV–compatible (with respect to A and
W) if the following
hold.
Ťk
‚ ρpGq Ď i“1 ξi pUi q.
‚ ρpAi q Ď ξi pUi q and η̄i ˝ ρæAi P Wi for each i.
Remark 6.12.
(1) We assume neither that ξi pUi q is algebraic nor that η̄i is a
polynomial map. The map η̄i merely makes the diagram commute.
(2) If (i) and (ii) are satisfied, we can force (iii) by setting
Uk`1 “ ξk`1 pUk`1 q “ Vk`1 “ G.
(3) If Vi ’s are not specified, then we will assume Vi “ t0u.
Example 6.13. Let us define a UV–structure on G “ PSL2 pRq by:
U1 “ SOp2q ˆ PSL2 pRq, V1 “ SOp2q, U2 “ exppRq ˆ PSL2 pRq, V2 “ exppRq,
and ηi px, yq “ x and ξi px, yq “ xy for i “ 1, 2. Note that ξ1 pU1 q is the set of elliptics,
and ξ2 pU2 q is that of hyperbolics. Then η̄1 , η̄2 are well-defined by
η̄1 pxq “ Rotp2π rotpxqq
and
η̄2 pxq “ exp `pxq.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

71

For a representation ρ P HompG, Gq, the set
η̄1 pρpGq X ξ1 pU1 qq
is the rotation spectrum of ρ. We may call
η̄2 pρpGq X ξ2 pU2 qq
as the (hyperbolic) length spectrum of ρ. The set of parabolics is
t1u Y PSL2 pRqz pξ1 pU1 q Y ξ2 pU2 qq ,
which is algebraic.
Then Lemma 4.23 generalizes as follows.
Lemma 6.14. Let us fix an algebraic group G equipped with a UV–structure, and
a finitely generated group G. Let A and W be as in Definition 6.11 and let
tρn : G Ñ Guně1
be a stably injective sequence of UV–compatible (with respect to A and W) representations. Suppose for each
i P t1, . . . , ku
and for each a P Ai , we have
η̄i ˝ ρm paq ‰ η̄i ˝ ρn paq
whenever m ‰ n. Then there exists a uncountable collection of faithful UV–
compatible (with respect to A and W) representations G Ñ G such that
tη̄i ˝ λpaq : λ P Λu
is uncountable for each i and a P Ai .
We omit the proof, which is an easy variation from Lemma 4.23.
Example 6.15. Let
H “ xa, b, t | rra, bs, ts “ 1y.
Recall the UV structure of PSL2 pRq in Example 6.13. In Example 4.25, we exhibited a stably injective sequence
φ p : H Ñ PSL2 pRq
such that
rot ˝φ p ptq “ 1{p
for p ě 2. We can further require that φ p paq is a hyperbolic element of length at
most 1{p by considering the moduli space of a torus with one cone point.
So, Lemma 6.14 implies that there exists a uncountable set
Λ Ď HompH, PSL2 pRqq

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S. KIM, T. KOBERDA, AND M. MJ

of faithful parabolic-free representations with all distinct rotation spectra and with
all distinct length spectra.
6.3. Combination Theorem for Smooth Actions. The last combination theorem
(Section 6.3) is one in the smooth category and its proof follows the lines of the
usual Baire category argument for the real line. Let Diff 8 pMq denote the group
of C 8 orientation preserving diffeomorphisms of a one-manifold M. Recall that
G ď Diff 8 pS 1 q is fully supported if the fixed point set of each element g P Gz1 has
empty interior. A group
G Ď Diff 8 pS 1 q
has a free orbit if G acts freely on some G–orbit. Following the strategy of [49,
Corollary 2.6] and [47], we will use the Baire category argument in the theorem
below. Note the similarity to Lemma 4.6.
Theorem 6.16. Suppose G and H are countable, fully supported subgroups of the
group Diff 8 pS 1 q. Then for a very general choice of
pψ, xq P Diff 8 pS 1 q ˆ S 1 ,
the group xG, H ψ y is isomorphic to G ˚ H and has xG, H ψ y ¨ x as a free orbit. In
particular, G ˚ H admits a faithful C 8 action on S 1 .
Theorem 6.16 seems to be a folklore theorem, the details of whose proof we
record here for the convenience of the reader. The theorem does not hold without
the hypothesis of being full supported, even for C 2 ; see [63].
In the proof below, for w P G ˚ H, we let }w} denote the word–length of w.
Proof. For ψ P Diff 8 pS 1 q, define ρψ : G ˚ H Ñ xG, H ψ y by the following:
ρψ æG “ IdG ,

ρψ phq “ hψ for h P H.

Let
X “ Diff 8 pS 1 q ˆ S 1 .
Since Diff 8 pS 1 q is a Frechét space [54], we see X is Baire. For each
w P G ˚ Hz1
we define a closed set
Xw “ tpψ, xq P X : ρψ pwqpxq “ xu.
By the Baire Category Theorem, it suffices to show the following.
Claim. Xw is nowhere dense for w ‰ 1.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

73

Let w be a shortest word such that Xw has a nonempty interior. Pick a nonempty
open set V Ď Xw . If
w “ g P Gz1,
then we have
Xg “ Diff 8 pS 1 q ˆ Fix g
is nowhere dense. If
w “ h P Hz1,
then we obtain
ď
(
tψu ˆ ψ´1 Fix h : ψ P Diff 8 pS 1 q
Xh “ tpψ, xq P X : ψ´1 hψpxq “ xu “
is nowhere dense. So, we assume that }w} ě 2.
For u, v P G ˚ H, we let
Yu,v “ tpψ, xq P X : ρψ puqρψ pvqpxq “ ρψ pvqpxqu.
There exists a continuous map Φv : Xu Ñ Yu,v defined by
pψ, xq ÞÑ pψ, ρψ pvq´1 pxqq.
As Φv´1 ˝ Φv is the identity, we have a homeomorphism Xu Ñ Yu,v . By minimality,
the set
ď
Y “ tYu,v : }u}, }v} ă }w}, and u ‰ 1u
is a countable union of closed nowhere dense sets, and hence has empty interior. So
V0 :“ VzY is a nonempty subset of Xw .
For each u P G ˚ H, we have
Xuwu´1 “ Yw,u´1 « Xw .
So, we may assume
ρψ pwq “ ψ´1 hk ψgk ¨ ¨ ¨ ψ´1 h1 ψg1
possibly after a conjugation. In particular, }w} “ 2k. Let us pick an arbitrary
pψ, x0 q P V0 . Define
x1 , . . . , x2k
by
x2i`1 “ gi px2i q
and by
x2i “ ψ´1 hi ψpx2i´1 q.
By assumption, x2k “ ρψ pwq.x0 “ x0 . Moreover, the points
x0 , x1 , . . . , x2k´1
are all distinct since V0 X Yu,v “ ∅ for }u}, }v} ă }w}.
Note that
ψpx0 q “ ψpx2k q “ hk ψpx2k´1 q.

74

S. KIM, T. KOBERDA, AND M. MJ

Choose a small closed interval J containing x0 such that xi R J for each 0 ă i ă 2k.
By perturbing ψ, we can pick pφ, x0 q P V such that
φæS 1 zJ “ ψæS 1 zJ
and
φpx0 q ‰ hk ψpx2k´1 q.
If i ă k, then
φ´1 hi φgi px2i´2 q “ φ´1 hi φpx2i´1 q “ φ´1 hi ψpx2i´1 q “ ψ´1 hi ψpx2i´1 q “ x2i
since hi ψpx2i´1 q R ψpJq and since
φ´1 æS 1 zψpJq “ ψ´1 æS 1 zψpJq .
Hence,
ρφ pwqpx0 q “

1
ź

φ hi φgi px0 q “
´1

2
ź

i“k
´1

φ´1 hi φgi px2 q “ ¨ ¨ ¨

i“k
´1

“ φ hk φgk px2k´2 q “ φ hk ψpx2k´1 q ‰ x0 .
So φ R Xw . We have a contradiction since we have assumed pφ, x0 q P V.



Theorem 6.16 implies that if G ď Diff 8 pS 1 q is fully supported then G ˚Z embeds
into Diff 8 pS 1 q as a subgroup admitting almost every point as a free orbit. One
actually has a genericity statement as below.
Corollary 6.17. Let G ď Diff 8 pS 1 q be countable and fully supported. Then for a
very general choice of
pψ, xq P Diff 8 pS 1 q ˆ S 1 ,
the group xG, ψy is isomorphic to G ˚ Z and has xG, ψy ¨ x as a free orbit.
Proof. Recall our notation Z “ xsy. If w P G ˚ Z, we let }w} denote the word length
in the following sense
}w} “ mint` : w “ s1 s2 ¨ ¨ ¨ s` ,

si P G Y tsu˘1 u.

For ψ P Diff 8 pS 1 q, define
ρψ : G ˚ xsy Ñ Diff 8 pS 1 q
by ρψ æG “ IdG and ρψ psq “ ψ. Then we can proceed very similarly to the proof of
Theorem 6.16, so we omit the details.


FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

75

7. Mapping Class Groups
In this section, we shift the focus to exotic actions of mapping class groups on
the circle, where here “exotic” means “not conjugate to Nielsen’s standard action”
(see [53, 29] for detailed discussions of Nielsen’s action). We first discuss actions
of fibered, hyperbolic 3–manifold groups on S 1 , in relation to Nielsen’s action.
7.1. The Universal Circle and Nielsen’s Action. Let us fix a marked point x P S g ,
and a distinguished lift x̃ of x in the universal cover H2 of S g .
For each homeomorphism φ of S g fixing x, we choose the preferred lift φ̃ of φ
to H2 which fixes x̃. There exists a continuous extension of φ̃ to BH2 « S 1 , which
depends only on the mapping class rφs P ModpS g , xq. The resulting action is called
Nielsen’s action and denoted as
νg : ModpS g , xq Ñ Homeo` pS 1 q.
Let us now pick a pseudo-Anosov homeomorphism Ψ on S g . We will denote the
mapping torus of S g with the monodromy map Ψ as
M “ S g r̂ Ψ S 1 .
Then M be a closed hyperbolic 3–manifold fibering over the circle, whose topological type is determined by the mapping class ψ “ rΨs P ModpS g q. We often abuse
the notation and use the mapping class ψ instead of the homeomorphism Ψ.
r of M is naturally homeomorphic to
The universal cover M
SĂg ˆ R – H2 ˆ R,
r by lines. There is a lamiwhere the R factor gives rise to a natural foliation of M
nation λ Ď S g preserved by Ψ, and λ gives a suspended codimension one foliation
Λ Ď M which is transverse to S g . This lamination in turn lifts to a lamination
rĎM
r which is transverse to SĂg .
Λ
The identification of SĂg with H2 gives rise to a natural faithful action of π1 pMq
on S 1 – BH2 . This action extends the given Fuchsian action of π1 pS g q on BH2 by
xψy – Z; see [40], for instance. This group xψy acts on S 1 with a finite even number
of fixed points (perhaps after replacing Ψ by a nonzero power), whose dynamics are
alternately attracting and repelling. For compactness of notation, this action will be
called the universal circle action of π1 pMq and denoted as
ρu : π1 pMq Ñ Homeo` pS 1 q.

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S. KIM, T. KOBERDA, AND M. MJ

We have a commutative diagram as below:
/ Z
_

π1 pM, xq

(7.1.1)

_

7

1

/ π1 pS g , xq

Push.

/

1ÞÑψ



ModpS g , xq
_

Fuchs.

'



/



ModpS g q

/

#

1

νg

Homeo` pS 1 q
Here, Push and Fuchs respectively denote the point pushing map and the prescribed
Fuchsian representation. Group theoretically, the middle row is isomorphic to
1 Ñ π1 pS g , xq Ñ Autpπ1 pS g , xqq Ñ Outpπ1 pS g , xqq Ñ 1.
The upper row is the split short exact sequence coming from the fibering. The
discussion in the previous paragraph shows that the square in the diagram is a fiber
product, all considered as subgroups of Homeo` pS 1 q. Hence we deduce the following, which is probably well–known (cf. [36], for instance, as well as [47, 25, 23]).
Proposition 7.1. The universal circle action ρu extends to Nielsen’s action νg .
Since the Fuchsian action of π1 pS g q on S 1 is minimal, so is the action νg . As an
element of π1 pMq, the mapping class ψ P ModpS g , xq will be identified with the
stable letter of the fibration of M over S 1 . In other words, we write
π1 pMq “ π1 pS g q ¸ xψy.
The conjugating action of the stable letter ψ on π1 pS g q comes from the Dehn–
Nielsen–Baer isomorphism, which says
ψ P ModpS g q – Outpπ1 pS g qq.
7.2. Exotic Mapping Class Group Actions. Let S 0 be a compact hyperbolic surface with nonempty geodesic boundary, and let x be a marked point in the interior of
S 0 . We put S 1 “ S 0 ztxu. In this section, we exhibit an inequivalent pair of faithful
representations
ModpS 0 , xq Ñ Homeo` pS 1 q.
We note that the mapping class group of a non-closed surface admits many other
interesting actions on R or S 1 , which we will not consider here. See [55, 14], for
instance.
Let us denote by p : S̃ 0 Ñ S 0 the universal cover of S 0 . We can identify S̃ 0 with
a convex subset of the hyperbolic plane. The argument in the previous subsection
also applies to S , and hence we have Nielsen’s action:
ν : ModpS 0 , xq Ñ Homeo` pS 1 q.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

77

This action is minimal on the limit set, which is a Cantor set. In particular, ν is not
semi-conjugate to the trivial action (Corollary 2.15).
On the other hand, we have Thurston’s faithful action
τ : ModpS 1 q Ñ Homeo` pRq.
This action is obtained by considering lifts of self-homeomorphisms of S 1 to the
universal cover S̃ 1 . Such lifts are chosen to fix one distinguished lift β̃ of β Ď BS 1 ,
so that ModpS 1 q acts on
pB S̃ 1 zβ̃q Y Bπ1 pS 1 , xq « R.
See [94, 47] for details, and [35] for related ideas. We have an embedding
i : Homeo` pRq Ñ Homeo` pS 1 q
as a subgroup with a global fixed point. Let τ̄ be the map obtained by composing
i ˝ τ with the isomorphism ModpS 0 , xq – ModpS 1 q. Since τ̄ is semi-conjugate to
the trivial action, we have the following.
Proposition 7.2. The faithful actions
ν, τ̄ ˝ α : ModpS 0 , xq Ñ Homeo` pS 1 q
are not semi-conjugate for all α P AutpModpS 0 , xqq.
In other words, ν and τ̄ do not merely fail to be semi-conjugate, but actually
inequivalent. In the spirit of Section 5, we ask:
Question 7.3. Do mapping class groups of (bounded or closed) compact hyperbolic
surfaces admit uncountably many inequivalent actions on the circle?
8. Zero rotation spectrum and Teichmüller theory
In this section, we consider free group and surface group actions on the circle,
and develop conditions under which the equivalence class of an action is determined by the rotation spectrum, and when the semi-conjugacy class of the action is
determined by the marked rotation spectrum.
In the case of indiscrete representations of groups into PSL2 pRq, there is a lack
of a geometric interpretation of such representations which is as well-developed
as Teichmüller theory in the case of discrete representations. In this section, we
consider the degree to which marked rotation spectrum can supplant marked length
spectrum as a (sometimes nearly complete) semi-conjugacy invariant.
Section 8.1 (see also Theorem 8.3) establish rigidity results. Here we conclude
that any two non-elementary faithful representations of a group into PSL2 pRq with
zero rotation spectrum are semi-conjugate (see also [74]). The remaining subsections focus more on flexibility results. We first describe properties of a linear (that
is, factoring through some finite–dimensional connected Lie group) action on the

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S. KIM, T. KOBERDA, AND M. MJ

circle in Section 8.2. In Sections 8.3, we analyze the circle actions of free and
surface subgroups of fibered hyperbolic 3–manifolds. In Section 8.4, we construct
smooth nonlinear actions of free groups.
8.1. Rigidity of Projective Actions. Let us gather some facts from Teichmüller
Theory which allow us to analyze projective actions of free and surface groups with
zero marked rotation spectrum.
Lemma 8.1. (cf. [73, 40]) Let S be an orientable surface and let
φ1 , φ2 : π1 pS q Ñ PSL2 pRq
be discrete and faithful representations of π1 pS q corresponding to complete finite
volume hyperbolic structures on S , with a fixed orientation. Then the corresponding
actions of φ1 and φ2 on S 1 are conjugate in Homeo` pS 1 q.
Proof. The two representations φ1 and φ2 of π1 pS q (considered up to conjugacy in
PSL2 pRq) correspond to two points X1 and X2 in the Teichmüller space T pS q or the
Teichmüller space T pS q with the opposite orientation. It suffices to show that if
X1 , X2 P T pS q,
then the corresponding representations are conjugate in Homeo` pS 1 q. Between any
two points in T pS q, there is a quasi-conformal map taking one hyperbolic structure
to the other. Writing h for such a map between X1 and X2 , we lift h to the universal
covers of X1 and X2 respectively. If S is a closed surface then h induces a quasi–
isometry r
h of the universal covers of X1 and X2 , which are both identified with H2 .
Thus, r
h induces a homeomorphism Br
h of S 1 which conjugates the actions of φ1 and
1
φ2 on S .
If S is not closed then h need not induce a quasi–isometry because S is noncompact. In this case, we consider h as a self-homeomorphism of S and note that
h induces a π1 pS q–equivariant homeomorphism between fundamental domains for
X1 and X2 in H2 . By cutting out small neighborhoods of the cusps of S , we obtain
a sequence of nested compact surfaces tS i uiPN whose union is S . Writing X1i for
the restriction of X1 to S i and X2i for the image of X1i under h, we get a sequence of
homeomorphisms
thi : X1i Ñ X2i uiPN
which lift to π1 pS q–equivariant homeomorphisms between fundamental domains
for X1i and X2i in H2 . Moreover, these homeomorphisms are compatible with respect
to inclusion, in the sense that if i ď j then h j “ hi when restricted to S i ; see [40,
Section 8.2.7] for more details.
The total preimages Ă
X1i and Ă
X2i of X1i and X2i in H2 (which are identified with the
corresponding universal covers) have boundaries which are naturally identified with

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

79

S 1 , after including the limit sets. In the limit as i tends to infinity, both boundaries
are identified with BH2 , and
h “ lim hi
iÑ8

induces a homeomorphism of S conjugating the two actions of π1 pS q.
1



Remark 8.2. When S is closed, representations corresponding to points in the two
different Teichmüller spaces T pS q and T pS q are not conjugate in Homeo` pS 1 q
because they have different Euler numbers.
Recall our convention that a surface group means the fundamental group of a
closed orientable hyperbolic surface. The following characterizes projective surface
group actions with rotation spectrum identically zero, up to conjugacy. The proof
is a fairly easy combination of standard facts from Teichmüller theory:
Theorem 8.3. If L is a finitely generated free group or a surface group, then there
exist only finitely many conjugacy classes of faithful projective actions φ of L with
rotation spectrum t0u. Such a representation φ is necessarily discrete.
Remark 8.4. Our proof below implies that there are exactly two conjugacy classes
if L is a surface group.
Proof of Theorem 8.3. Let us first assume L is a surface group. The fact that φpLq is
a discrete cocompact subgroup of PSL2 pRq is an immediate consequence of Lemma
2.24. By Lemma 8.1, we have that two discrete faithful surface group representations into PSL2 pRq with equal Euler numbers give rise to conjugate actions on S 1 .
The Euler number of such a representation is a conjugacy invariant in Homeo` pS 1 q,
and it takes on exactly two values for discrete surface group representations [48].
We now assume L is free. We perform much of the same analysis as for surface
groups. As the rank–one case is trivial, we suppose L is non-elementary. Let φ is a
representation in
Φ “ tφ P Hom pL, PSL2 pRqq : rot ˝φpLq “ t0u and φ is faithfulu.
Lemma 2.24 implies the discreteness of φ. In particular, S “ H2 {φpLq is a noncompact hyperbolic surface with cusps or funnels (i.e. flaring ends). Denote by
S 0 the (compact) convex core of S , and S 1 be the double of S 0 along the geodesic
boundary components.
Since we have
χpS q “ 1 ´ rankpLq,
there are finitely many possible homeomorphism types of the surface S 1 . Each
homeomorphism type corresponds to at most two conjugacy classes (depending on
the choice of the orientation) of actions by Lemma 8.1. In other words, if φ1 denotes
the Fuchsian representation of π1 pS 1 q determined by φ, then there are only finitely

80

S. KIM, T. KOBERDA, AND M. MJ

many possible conjugacy classes of φ1 in Homeo` pS 1 q. As φ is a restriction of φ1 ,
there are only finitely many conjugacy classes in Φ.

Question 1.20 from the introduction asks to what degree marked spectrum generally controls the equivalence class of a projective surface group action. In light
of Theorem 8.3 it seems likely that there should be many inequivalent projective
actions with a fixed nonzero marked rotation spectrum.
Question 8.5. How many semi-conjugacy classes of projective free group actions
are there on the circle with a given nonzero unmarked rotation spectrum?
A marked rotation spectrum of a free group determines the semi-conjugacy class
of an action. The discrete case follows from the classification of Fuchsian actions,
and the indiscrete case is proved in [104].
8.2. Lie Subgroups of the Circle Homeomorphism Group. In order to describe
precisely what we mean by “exotic” actions of free and surface groups, let us make
the following definition.
Definition 8.6. Let L be a group. A representation
ρ : L Ñ Homeo` pS 1 q
is said to be linear if there exists a finite–dimensional connected Lie group G ď
Homeo` pS 1 q such that ρ factors through G.
If G is a group acting on a space X, we define its support as
ď
supp G “
supp g “ tx P X : gpxq ‰ x for some g P Gu.
gPG

Lie groups in Homeo` pS 1 q can be characterized as follows.
Theorem 8.7 ([47, Section 4]; see also [74]). If G is a finite–dimensional connected
Lie group in Homeo` pS 1 q, then one of the following holds:
pkq

(i) supp G “ S 1 , and G is conjugate to either SOp2, Rq or PSL2 pRq for some
1 ď k ă 8;
(ii) supp G ‰ S 1 , and G is isomorphic to a subdirect product of copies of PSL„
2 pRq,
R and AffpRq.
The following consequence is what we will use.
Lemma 8.8. Let L “ xs1 , . . . , sm y be a finitely generated group, and let ρ : L Ñ
Homeo` pS 1 q be a representation. Suppose ρ is semi-conjugate to
ρ1 : L Ñ G ď Homeo` pS 1 q
for some finite–dimensional connected Lie group G. If Fix ρpLq “ ∅, and if
pkq
Fix ρpsi q ‰ ∅ for each i, then G is conjugate to PSL2 pRq for some 1 ď k ă 8.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

81

Proof. Since the existence of a global fixed point is a semi-conjugacy invariant
property (Corollary 2.15), we have that Fix ρ1 pLq “ ∅. In particular, we have the
alternative (i) of Theorem 8.7 for G.
Note that rot ˝ρ1 psi q “ rot ˝ρpsi q “ 0. If G is conjugate to SOp2, Rq, then we
would have that ρ1 psi q “ 1 and supp ρ1 pLq “ ∅. Hence, G must be conjugate to
pkq
PSL2 for some 1 ď k ă 8.

pkq

Let us investigate actions of PSL2 pRq in more detail. We fix 1 ď k ă 8. Denote
by pk : R{kZ Ñ R{Z “ S 1 the natural k–fold cover. There exists a k–fold cover
pkq

qk : Homeo` pS 1 q Ñ Homeo` pS 1 q
pkq

such that the following diagram commutes for all g P Homeo` pS 1 q:
g

R{kZ


/

R{kZ

pk
qk pgq

S1

/



pk

S1

More concretely, we can write
pkq

Homeo` pS 1 q “ tg P Homeo` pR{kZq : gpx ` 1q “ gpxq ` 1u.
pkq

Note that PSL2 pRq is the fiber product given by the following commutative diagram, where the lower row is a central extension:
/

pkq

(8.2.1)

PSL2 pRq
9

1

/

Z{kZ

/


pkq
Homeo` pS 1 q

qk

/

PSL2 pRq


Homeo` pS 1 q

$/

1

The following is immediate from definition.
pkq

Lemma 8.9. For each g P Homeo` pS 1 q, we have
rot ˝qk pgq “ k rot g

mod Z.

Let us describe the limit sets of lifts of projective actions.
Lemma 8.10. For a finitely generated group L ď Homeo` pS 1 q and for 1 ď k ă 8,
we have
´1
Λ ˝ q´1
k pLq “ pk ˝ ΛpLq.
Proof. If L has a finite orbit, then both sides are empty. So we assume ΛpLq ‰ ∅.
´1
Note that p´1
k ˝ ΛpLq is nonempty and qk pLq–invariant. By minimality, we have
´1
Λ ˝ q´1
k pLq Ď pk ˝ ΛpLq.

82

S. KIM, T. KOBERDA, AND M. MJ

To show the opposite inclusion, let us pick arbitrary
x P p´1
k ˝ ΛpLq,

y P R{kZ.

There exists a sequence tgn u Ď L such that gn ˝ pk pyq Ñ pk pxq. Write gn “ qk phn q
for some hn P q´1
k pLq. We have qk phn q ˝ pk pyq “ pk ˝ hn pyq. Hence possibly after
postcomposing hn with suitable 2πZ{k rotations, we have that hn pyq Ñ x. This
implies x is in the closure of q´1

k pLqpyq, as desired.
Lemma 8.11. Let L be a finitely generated group, and let 1 ď k ă 8. If there
exists a commutative diagram
pkq

Homeo` pS 1 q
ρ

6

ρ0

L

/



qk

Homeo` pS 1 q

then we have
Λpρq “ p´1
k ˝ Λpρ0 q.
Proof. From the central extension (8.2.1), we see that
“ ´1
‰
qk ˝ ρ0 pLq : ρpLq ď k ă 8.
By Lemma 2.8, we obtain
Λ ˝ ρpLq “ Λ ˝ q´1
k ˝ ρ0 pLq.
The desired conclusion follows from Lemma 8.10.



We will need the following analogue of Lemma 2.23.
Lemma 8.12. Let 1 ď k ă 8, and let L be a finitely generated group. Suppose we
have an action ρ of L on R{kZ such that
ρ : L Ñ PSL2 pRq ď Homeo` pR{kZq.
pkq

Then a minimalization of ρ can be chosen as some representation
ρ̄ : L Ñ PSL2 pRq.
pkq

Note that a priori, the minimalization ρ̄ of ρ merely maps into Homeo` pR{kZq,
pkq
and not necessarily into PSL2 pRq.
Proof of Lemma 8.12. We may assume ρ is faithful, by considering L{ ker ρ instead
of L if necessary. If ρ is minimal, then we have nothing to prove. If ρ has a finite
orbit then ρ̄ factors through SOp2, Rq, which can be also regarded as a subgroup of
pkq
PSL2 pRq. So we will assume that Λpρq is a Cantor set.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

83

We put ρ0 “ qk ˝ ρ. By Lemma 8.11, we have
Λpρq “ p´1
k ˝ Λpρ0 q.
Note that ρ0 pLq is discrete; otherwise, ρ0 pLq would contain an irrational rotation
by Lemma 2.24 and ρ would be minimal. The same reasoning shows that ρ0 pLq is
non-elementary. We see that Λpρ0 q is a Cantor set.
By Lemma 2.23, the minimalization ρ̄0 of ρ0 can be chosen as a finite-area Fuchsian representation of
L{ ker ρ0 – ρ0 pLq.
There is a Cantor function h0 : S 1 Ñ S 1 for Λ ˝ ρ0 pLq such that
ρ0 ěh0 ρ̄0 .
We claim that there exists a Cantor function h on R{kZ and a representation
ρ̄ : L Ñ Homeo` pR{kZq such that the cube diagram in Figure 1 holds for all g P L,
where all the vertical maps denote pk . From the standard covering theory for the
map pk , we can define h so that the front, left, right and bottom faces all commute.
Then h is the Cantor function corresponding to Λpρq. If hpxq “ hpyq for some
x, y P R{kZ, then h ˝ ρpgqpxq “ h ˝ ρpgqpyq; this follows from that Λpρq is ρpLq–
invariant. So there exists a well-defined map ρ̄ such that the upper face of the cube
commutes. The rear face commutes by diagram chasing.
We have ρ̄ ě pk ρ̄0 from the cube. In particular, we see
ρ̄ P HompL, PSL2 pRqq.
pkq

We conclude that ρ̄ is minimal from Lemma 8.11 and from that
Λpρ̄q “ p´1
k ˝ Λpρ̄0 q “ R{kZ.
ρ̄pgq

R{kZ
h

;

h
ρpgq

R{kZ

/



ρ̄0 pgq

h0

S

1

R{kZ
;

R{kZ

1
:S



/



/



1
:S

h0
ρ0 pgq

/



S1

Figure 1. Proof of Lemma 8.12. All the vertical maps are pk .

84

S. KIM, T. KOBERDA, AND M. MJ

Remark 8.13. Let g P Homeo` pS 1 q, and let x P S 1 be an isolated fixed point of
g. We say x is a hyperbolic fixed point if x is an attracting or repelling fixed point;
otherwise, we say x is a parabolic fixed point; see for example [11]. Note also that a
pkq
non-elliptic element h P PSL2 pRq may or may not have a fixed point; however, hk
will have k fixed points which are parabolic, or 2k fixed points which are hyperbolic.
So in the above lemma, for each g P Lz ker ρ̄ the fixed point set of ρ̄pgq on R{kZ are
all hyperbolic or all parabolic.
Let us now note easy constructions of circle actions by free and surface groups
which are not projective even up to semi-conjugacy. Hence it makes more sense to
quest a free or surface group representation that is not conjugate or semi-conjugate
to any linear action in the sense of Definition 8.6. We begin with a general observation:
Example 8.14. Fix a positive divisor k ą 1 of 2g ´ 2. Each Fuchsian representation
ρ : π1 pS g q Ñ PSL2 pRq ď Homeo` pS 1 q
has the Euler number epρq “ ˘p2g ´ 2q, which is the translation number of the lift
of the relator in π1 pS g q to R. As k divides 2g ´ 2, this R–action projects to some
faithful R{kZ–action
ρ1 : π1 pS g q Ñ PSL2 pRq ď Homeo` pR{kZq.
pkq

By Lemma 8.9, the rotation spectrum is ρ1 is contained in Q{Z. Also, the action ρ1
is minimal by Lemma 8.11. Assume that ρ1 is semi-conjugate to some projective
action. Then Lemmas 2.24 and 2.23 would imply that ρ1 is conjugate to some
discrete projective action. This is a contradiction, since the k–th power of every
nontrivial element fixes exactly 2k points under ρ1 . We conclude that ρ1 is not semiconjugate to any projective action.
Example 8.15. We can lift a finite–area Fuchsian representation ρ of Fn to:
ρ : Fn Ñ PSL2 pRq ď Homeo` pR{kZq.
pkq

The same argument as above shows that ρ is not semi-conjugate to any projective
action for k ą 1.
8.3. Free and surface subgroups of π1 pMq. Throughout this subsection, we let M
be a closed fibered hyperbolic 3–manifold
˜ tS 1
M “ S rˆ
for some genus r ě 2 and some pseudo-Anosov mapping class t on S g ; see (7.1.1).
We have the universal circle action and the Kleinian action:
ρu : π1 pMq Ñ Homeo` pS 1 q,
ρK : π1 pMq Ñ PSL2 pCq ď Homeo` pS 2 q.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

85

We will make two additional assumptions:
(i) ρu ptq fixes at least four points;
(ii) ρu paq fixes at least one point for each a P π1 pMq.
The assumption (i) simply means that a stable foliation of t has no single-prong
singularity [29]. The assumption (ii) can always be guaranteed after taking a finite–
sheeted cover of M; see Lemma 8.16.
We will study the restriction of ρu on a surface subgroup G ď π1 pMq. In the case
when G is quasi-Fuchsian, we will show that ρu æG is semi-conjugate to a Fuchsian
actions. When G is geometrically infinite, we sketch K. Mann’s argument [71] that
ρu æG is “often” not projective even up to semi-conjuguacy.
For existential results, we find a quasi-Fuchsian surface group G in π1 pMq such
that ρu æG is not conjugate to any linear actions. We will also find a finitely generated
free subgroup F ď π1 pMq such that ρu æF is non-linear even up to semi-conjugacy.
All of the above actions will have rotation spectra identically zero (by the second
assumption), and hence will share some of the features of discrete projective surface
group actions.
8.3.1. Quasi-Fuchsian Surface Subgroups. We first justify the assumption (ii):
Lemma 8.16. Let N be a closed fibered hyperbolic 3–manifold, and let ρ denote
the universal circle action of N. Then there exists a finite sheeted cover N 1 of N
such that rot ˝ρpπ1 pN 1 qq “ t0u.
Proof. We will denote ρpgqpxq “ g.x for g P π1 pNq and x P S 1 . Let t denote the
stable letter of π1 pNq from the given fibering. Then for some L ą 0, the pseudoAnosov element t L has an even number of fixed points, in such a way that attracting
and repelling fixed points alternate [29]. Define a finite sheeted cover N 1 Ñ N by
π1 pN 1 q “ π1 pS r q ¸ xt L y.
We claim every element of π1 pN 1 q fixes at least one point. Consider an arbitrary
ht P π1 pN 1 q. For brevity we assume k ą 0, as the case k ă 0 is similar. We may
only consider the case that Fix h and Fix t L are disjoint. The hyperbolic element
h P PSL2 pRq has one attracting and one repelling fixed points on S 1 . Pick an
interval J “ pp, qq on S 1 such that p and q are attracting fixed points of h and of
t L , respectively. We choose J to be innermost, in the sense that J contain no fixed
points of h or t L . Up to reversing, we have a circular ordering
Lk

p ă ht Lk .p ă ht Lk .q ă q.
This implies ht Lk fixes some point in J, as desired.
The first main point of this subsection is the following.



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Theorem 8.17. If G is a quasi-Fuchsian surface subgroup of π1 pMq, then ρu æG is
semi-conjugate to a Fuchsian action of G.
Proof. Let H “ G X π1 pS r q, the intersection of G with the fiber subgroup. Limit
sets in S 1 and S 2 will be denoted by superscripts 1, 2 respectively. We have:
‚ ΛG1 “ Λ1H “ pρu pHq.xq1 Ď S 1 for any x P S 1 ;
‚ ΛG2 “ Λ2H “ pρK pHq.xq1 Ď S 2 for any x P S 2 .
The Cannon-Thurston map CT maps Λ1H continuously and H´equivariantly onto
Λ2H ([27]). Since attracting fixed points of elements of H are dense in both Λ1H and
Λ2H , it follows that Λ1H Ă CT ´1 pΛ2H q.
There are now two natural circle actions of H (and hence G, since H is normal in
G with quotient Z):
(1) The minimalization of the action of H on S 1 obtained by collapsing the
complementary arcs of Λ1H . Thus, the minimalization is a standard semiconjugacy and one gets an action of G on S 1 as a result;
(2) The action of H on Λ2H “ ΛG2 , which is homeomorphic to S 1 . Since ΛG2 is
homeomorphic to a circle and the G´action on Λ2H is conjugate to the usual
Fuchsian action of G on S 1 , we obtain another action of G on S 1 through
the Cannon-Thurston map.
So, it suffices to prove that these two actions of G on S 1 are topologically conjugate, i.e. the Cannon-Thurston map CT is (topologically) identical to the minimalization. Pairs of points on the boundary of S 1 zΛ1H will be referred to as adjacent
boundary points. To prove that CT is (topologically) identical to the minimalization
map it suffices to prove that CT preserves circular order, or equivalently:
Claim 1: Adjacent boundary points of Λ1H are identified by CT ;
Claim 2: No other pair of points of Λ1H are identified by CT .
First for any pair of points p, q P H2 Y BH2 draw the (bi–infinite or semi–infinite
or finite) geodesic λ joining them in H2 , the universal cover of the fiber S r . Flow λ
in both directions in M̃ under the natural foliation by lines induced by the pseudoAnosov monodromy. The full flow image Lλ is called the ladder corresponding to
λ in [81, 82]. The ladder Lλ has the following properties:
(1) Lλ is uniformly (which is to say independently of λ) quasiconvex [81,
Lemma 4.1]; [81, Theorem 3.8].
(2) Let Λ Ă BH2 be a closed subset invariant under the action of the pseudoAnosov monodromy, and let HΛ denote the convex hull of Λ. Suppose λ
is a geodesic segment in HΛ , each of whose endpoints lie either in Λ or on
a boundary geodesic of HΛ , and which is not a boundary geodesic of HΛ .
Then the ladder Lλ separates the full flow images of the two halves of HΛ
separated by λ (see [80, Section 4.2] and [83, Proposition 4.5]).

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

87

It follows that if λ is bi–infinite but not a boundary bi–infinite geodesic of the
convex hull of Λ1H (or equivalently that the end-points p and q of the bi–infinite
λ are not adjacent points of Λ1H ), then CT ppq ‰ CT pqq; otherwise the common
singleton image-point CT ppq “ CT pqq would separate CT pΛ1H q “ Λ2H , where the
latter is homeomorphic to S 1 . This proves Claim 2.
To prove Claim 1, let p and q be adjacent boundary points and let λ be the
boundary bi-infinite geodesic of the convex hull of Λ1H joining p and q. Suppose
CT ppq ‰ CT pqq. Choose a point z on λ, join it to a point w P Λ1H other than p and
q, thus obtaining a geodesic ray µ. Flow µ to get the ladder Lµ separating p from
q. Then the limit set of Lµ is the singleton CT pwq in Λ2H , though it separates this
latter limit set, which is a contradiction. This proves Claim 1.

We address the second main point of Section 8.3.
Theorem 8.18. There exists a quasi-Fuchsian surface subgroup G ď π1 pMq such
that ρu æG is not conjugate to any (possibly non-injective) linear action.
We will employ the following consequence of [60]. We are grateful to J. Kahn
for explaining the relevant part to us in the argument below.
Lemma 8.19 (See [60]). For each element g P π1 pMq, there exists an N " 0 and a
quasi-Fuchsian surface subgroup G ď π1 pMq such that gN P G.
Proof. The construction of Kahn and Markovic in [60] proceeds by gluing pairs of
pants together to construct closed surface subgroups G ď π1 pMq. This is arranged
in such a way that for a given  ą 0 and for R ą 0 sufficiently large, if the translation
length `phq of h P π1 pMq satisfies |`phq´R| ď {2, then a surface group G ď π1 pMq
can be required to contain h; see also [59, Section 4.1]. In particular, we can require
that gN P G for some N " 0. We remark that geometrically, the group G is forced
to be quasi–Fuchsian (see also [31]).

Proof of Theorem 8.18. By Lemma 8.19, there exists a quasi–Fuchsian surface subgroup G ď π1 pMq that contains t N for some N " 0. Assume for contradiction that
ρ :“ ρu æG is conjugate to a linear action. Write H “ G X π1 pS r q E G. Since
G{H ď π1 pMq{π1 pS r q – Z,
and since ρu restricts to a Fuchsian action on π1 pS r q, the restriction ρæH is a nonelementary Fuchsian action. In particular, ρ has no global fixed point. By Lemma 8.8,
for some integer k ě 1 we may write
ρu : G Ñ PSL2 pRq.
pkq

We have k ą 1 since t N P G. On the other hand, each ρpuq P ρpHqzt1u has exactly
two hyperbolic points on S 1 . This contradicts Remark 8.13.


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S. KIM, T. KOBERDA, AND M. MJ

Recall [27] that the Cannon-Thurston map identifies precisely end-points of stable and unstable laminations (and ideal points of complementary polygons). It follows that the convex hull of
Λ :“ Λ ˝ ρu pGq “ Λ ˝ ρu pHq Ď BH2
is an infinite sided polygon whose sides are leaves of the stable and unstable laminations. Such a polygon is referred to as a leaf-polygon in [31]. In [31, Proposition
3.9], it is shown that if G corresponds to a quasi-Fuchsian surface transverse to the
flow, then the convex hull of Λ1H is a leaf-polygon. We thus have the following
generalization:
Corollary 8.20. If G ď π1 pMq be a quasi-Fuchsian surface subgroup, then the
convex hull of the limit set of G X π1 pS r q in S̃ p“ H2 q is a leaf-polygon.
8.3.2. Geometrically infinite surface groups. In regard of Theorems 8.17 and 8.18,
the following question suggests itself:
Question 8.21. For which surface subgroups G ď π1 pMq, is the restriction ρu æG
semi-conjugate to a linear action?
By Theorem 8.17, such a surface group G is necessarily geometrically infinite.
The difficulty of repeating the argument of Theorem 8.18 in addressing Question 8.21 has to do with the location of the limit set of a certain subgroup of the
fiber group. Indeed, let G ď π1 pMq be a surface subgroup containing t N for some
N " 0, as in the proof of Theorem 8.18. Put H “ G X π1 pS r q and write
Λ :“ Λ ˝ ρu pGq “ Λ ˝ ρu pHq.
The set Λ in general will be smaller than the limit set of π1 pS r q, since H is not
normal in π1 pS r q. It is possible that the entirety of Λ lies in between two consecutive
attracting points of t. See Figure 2.
If Λ is entirely contained between two attracting fixed points of the stable letter,
then the minimalization of the action of the action of G may collapse the fixed
points of t into two or one fixed point, which is dynamically indistinguishable from
a hyperbolic or parabolic fixed point. Note that in this situation, Λ contains at most
three fixed points of the stable letter t. However, we have the following proposition:
Proposition 8.22. Let G ď π1 pMq be a surface subgroup containing t N for some
N ą 0. Suppose that the limit set Λ of H “ G X π1 pS r q contains more than four
fixed points of ρ0 ptq. Then the action of G is not semi-conjugate to any (possibly
non–injective) linear action.
Proof. Note that the limit set Λ is perfect, and the minimalization map Λ Ñ S 1
is at most two–to–one. It follows that if Λ contains more than four fixed points
of the stable letter t then the action of t in the minimalization will have at least

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

89

Figure 2. The stable letter stabilizes the blue and the red geodesics,
corresponding to attracting and repelling fixed points of the stable
letter, respectively. The green dotted geodesics connect points in the
limit set Λ of H.
three fixed points, whereas each element of H will have either one or two fixed
points by Lemma 2.23. By the same argument as Theorem 8.18, we see that such
a minimal action cannot be conjugate (and hence cannot be semi-conjugate) to any
linear action.

Let us now assume G ď π1 pMq is a geometrically infinite surface subgroup.
From a result by Bonahon and Thurston [10], we have a finite cover M 1 Ñ M such
that G is the associated fiber subgroup of some fibering of M 1 . We say G is in the
original fibered face, if G and the restriction of π1 pS r q belongs to the same fibered
face of the Thurston norm sphere for M 1 , possibly up to multiplication by ´1.
We can now state the third main point of Section 8.3.
Theorem 8.23. Let G be a fiber subgroup of π1 pMq associated to some fibering
over S 1 . If G is in the original fibered face, then ρu pGq is conjugate to a Fuchsian
action.
Proof. Theorem 14.11 of [90] shows that a closed embedded geometrically infinite
fiber surface in M can be isotoped to be transverse to the flow if and only if it lies in
the fibered face of the Thurston unit norm sphere. Let F be any such surface with
fundamental group GpĂ π1 pMqq. Then G acts on the leaf space of the flow (since
F is transverse to the flow). Since both Srr and F̃ are sections of the flow (S r being
the fiber) on M̃, the G actions on Srr and F̃ are conjugate. Hence the universal circle
action of G is conjugate to the action of G on B F̃, which is the standard action. 

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S. KIM, T. KOBERDA, AND M. MJ

As we remarked in the introduction, K. Mann has a construction of surface group
actions on S 1 that is not semi-conjugate to any faithful projective action, which
appears in her thesis [71]. In fact, she showed that if a fiber subgroup G ď π1 pMq
is not in the original fibered face, then ρu æG is not semi-conjugate to any faithful
projective action. Let us sketch her argument.
Note that the Thurston norm is the dual to the Euler class of the surface group
fiber; see [98]. Since G E π1 pMq, we have
Λ ˝ ρu pGq “ Λ ˝ ρu pMq “ S 1 .
In other words, ρu æG is minimal. If ρu æG is semi-conjugate to a projective action,
then one can find
ρ : G Ñ PSL2 pRq
which is conjugate to ρu æG . As rot ˝ρu pGq “ t0u, we see ρ is actually Fuchsian
(Lemma 2.24). In particular, ρ˚ eu is a maximal Euler class. But this contradicts the
fact that the surface subgroup came from a different fibered face of the Thurston
norm ball.
8.3.3. Free subgroups of π1 pMq. Let us now show that the universal circle action
restricts to some non-linear free group action, in the spirit of Theorem 8.18. We
recall the following straightforward fact, which follows from an easy ping–pong
argument; see for instance [17, p. 467].
Lemma 8.24. Let G be a torsion-free word-hyperbolic group, and let tg1 , . . . , gn u Ď
G be elements such that no two of them generate a cyclic group. Then for all
sufficiently large N ą 0, the group xg1N , . . . , gnN y is free of rank n.
The last main point of Section 8.3 is the following.
Theorem 8.25. There exists a finitely generated free subgroup F of π1 pMq such that
ρu æF is not semi–conjugate into any linear action of F.
Proof. Since π1 pS r q acts minimally, we can choose a minimal collection of elements X Ď π1 pS r q such that each component of
S 1 z Fix ρu ptq
contains a fixed point of some element in ρu pXq. We set F to be the free group
generated by sufficiently high powers of t and of the elements of the finite set X, as
guaranteed by Lemma 8.24. Say t N P F for some N ą 0, and put ρ “ ρu æF .
Write H “ F X π1 pS r q E F. Since
F{H ď π1 pMq{π1 pS r q – Z,
the restriction ρæH is a non-elementary Fuchsian action. The limit set
Λ :“ Λ ˝ ρpFq “ Λ ˝ ρpHq

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

91

contains Fix ρptq and Fix ρpaq for all a P H.
Assume for contradiction that ρ is semi-conjugate to a linear action ρ̄. By Lemma 8.8,
there exists some integer k ě 1 such that
ρ̄ : F Ñ PSL2 pRq.
pkq

By Lemma 8.12, we may assume ρ̄ is a minimalization of ρ. In particular, there
exists a surjective monotone degree one map h on S 1 such that
ρ ěh ρ̄.
We claim now that no two fixed points of t are identified under h. Indeed, h
is either two–to–one or one–to–one at every point in the limit set Λ. Precisely, h
is either a homeomorphism or given by the “devil’s staircase map” or “Nielsen’s
Cannon–Thurston map” (see [41]). Since any interval bounded by two fixed points
of ρptq contains a point in Λ within its interior, we have that h cannot identify
distinct fixed points of t, thus establishing the claim. It follows that ρ̄ptq fixes at
least four points and so, k ě 2.
Since Λ “ Λ ˝ ρpHq, we see that ρ̄æH is a minimalization of ρæH . Let a P H.
By Remark 8.13, the set Fix ρ̄paq consists of one parabolic or two hyperbolic fixed
pkq
points. This contradicts that ρ̄pFq ď PSL2 pRq.

8.4. Nonlinear Smooth Actions of Free Groups. So far, we have described nonlinear actions of free groups regarded as subgroups of fibered hyperbolic 3–manifolds,
which in turn are regarded as subgroups of mapping class groups. However, Nielsen’s
actions of mapping class groups are not smooth. Actually, no finite index subgroups
of mapping class groups admit C 2 –smooth actions on the circle [3].
We will now produce nonlinear actions of free groups that are C 8 –smooth.
Theorem 8.26. For each n ě 2, there exists a faithful action
1
ρn : Fn Ñ Diff 8
` pS q

which is not semi-conjugate to any linear action of Fn on S 1 .
Recall the definition of hyperbolic and parabolic fixed points from Remark 8.13.
Lemma 8.27. Let g, ḡ P Homeo` pS 1 q, and let h : S 1 Ñ S 1 be a surjective monotone degree one map such that h ˝ g “ ḡ ˝ h. If h is injective on Fix g, then we
have:
(1) Fix ḡ “ hpFix gq;
(2) If x P Fix g is a hyperbolic fixed point, then so is hpxq;
(3) If x P Fix g is a parabolic fixed point, then so is hpxq.

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S. KIM, T. KOBERDA, AND M. MJ

Proof. For part (1), the inclusion hpFix gq Ď Fix ḡ is obvious. To show the opposite
inclusion, assume for some x P S 1 z Fix g that
h ˝ gpxq “ ḡ ˝ hpxq “ hpxq.
Then J “ h´1 phpxqq is a non-degenerate closed interval containing x and gpxq.
Since xgy acts on J, we see that BJ Ď Fix g and that hpBJq “ thpxqu. This contradicts the assumption that hæFix g is injective. So, part (1) is proved.
The above proof shows that for each y P Fix ḡ the set h´1 pyq is a singleton. For
each isolated fixed point x P Fix g, the point hpxq is an isolated point in Fix ḡ. If V
is an open neighborhood of hpxq such that
Fix ḡ X V “ thpxqu,
then the only fixed point of g in U “ h´1 pVq is x. By looking at the dynamics of
the group xgy on U, we obtain parts (2) and (3).

Proof of Theorem 8.26. Let us first consider the case n “ 2. We put
A “ t0, 1{3, 2{3u Ď S 1 .
1
Pick a P Diff 8
` pS q such that Fix a “ A and such that
$
’
&pt, 1{3q, if t P p0, 1{3q,
aptq P p1{3, tq, if t P p1{3, 2{3q,
’
%p2{3, tq, if t P p2{3, 1q.

The fixed points 0 and 1{3 are hyperbolic and 2{3 is parabolic; see Figure 3.
1
3

a
νp0q
0
b “ aν

2
3

Figure 3. Proof of Theorem 8.26.
1
ν
By Theorem 6.16, there is ν P Diff 8
` pS q such that xa, a y – F 2 and moreover,

(8.4.1)

t ă νptq ă t ` 1{3 for all t P A.

This is because (8.4.1) is an open condition for ν. Let us denote a generating set of
1
F2 as ta0 , b0 u, and define ρ : F2 Ñ Diff 8
` pS q by
ρpa0 q “ a,

ρpb0 q “ aν .

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

93

Let ρ̄ be the minimalization of ρ, and let h : S 1 Ñ S 1 be a surjective monotone
degree one map so that
ρ ěh ρ̄,
as in Definition 2.9. By construction, the finite set
C :“ A Y νpAq
is contained in closure of every orbit of ρ; this follows from that the supporting
intervals of a and aν form a “chain” of intervals, as defined in [56]. We see that
Λpρq is infinite and contains C.
So, the map h is either two-to-one or one-to-one at each point of C. Since A
and νpAq are alternating on S 1 and since h is monotone degree–one, we see that
hæA is injective. By Lemma 8.27, the set hpAq will contain both hyperbolic and
parabolic fixed points of ρ̄paq. Lemma 8.12 implies that ρ̄ is not conjugate to the
pkq
minimalization of a representation of F2 into PSL2 pRq. The proof is now complete
by Lemma 8.8.
For the case n ą 2, we simply consider the embedding as a finite index subgroup
Fn Ñ F2
and define ρn as the restriction of ρ on Fn . Then for some N ą 0, we have
xaN , paν qN y ď ρpFn q.
Since all the dynamical properties of xa, aν y that we used are shared by xaN , paν qN y
as well, we see that ρn is not linear even up to semi-conjugacy.

Appendix A. Equivalent Notions of Semi-Conjugacy
The aim of this appendix is to illustrate the equivalence of the different notions
of semi-conjugacy in Theorem 2.12. We will mainly give an account of the fact that
monotone equivalence and minimalization equivalence both coincide with the other
notions of semi-conjugacy. We will also guide the reader through the literature for
a proof that the rest of the notions of semi-conjugacy are equivalent to each other.
The facts in this section are mostly based on Ghys’ original ideas in [45, 47].
Readers are also referred to very recent surveys by Mann [73] and by Bucher–
Frigerio–Hartnick [19].
Throughout this appendix, we let L be a countable group.
Constructing a common blow-up
The implication of the following lemma is that the ě–relation on R induced by a
(possibly discontinuous) semi-conjugating map can be “generated” by continuous
semi-conjugating maps.
Lemma A.1. Let L be a countable group.

94

S. KIM, T. KOBERDA, AND M. MJ

(1) Suppose we have two actions
ρ0 , ρ1 P HompL, Homeo` pRqq.
Then ρ0 ěh ρ1 for some proper nondecreasing map h if and only if there
exists an action ρ P HompL, Homeo` pRqq and surjective nondecreasing
maps
h1 , h2 : R Ñ R
such that ρ ěhi ρi for i “ 1, 2.
(2) Suppose we have two actions
ρ0 , ρ1 P HompL, HomeoZ pRqq.
Then ρ0 ěh ρ1 for some monotone degree one map h if and only if there
exists an action ρ P HompL, HomeoZ pRqq and surjective monotone degree
one maps
h1 , h2 : R Ñ R
such that ρ ěhi ρi for i “ 1, 2.
Proof. Let us prove part (2) of the lemma, since the proof of part (1) is very similar
and simpler. We first consider the forward direction. For each x P R, let
hpx´q “ lim hptq
tÑx´0

and hpx`q “ lim hptq.
tÑx`0

Let us consider the strip
X “ R ˆ I Ď R2 .
For two points p, q P X, we denote by rp, qs the segment joining p and q. Define a
singular foliation of X:
F “ trpx, 0q, py, 1qs : x P R and y P rhpx´q, hpx`qsu .
Each point px, 1{2q P X belongs to a unique leaf, say Lpxq P F . So there exists
maps hi : R Ñ R uniquely determined by the condition:
Lpxq “ rph0 pxq, 0q, ph1 pxq, 1qs.
The map hi is monotone, since leaves do not intersect in the interior of X. Also
h˝T “T ˝h
implies that
hi ˝ T “ T ˝ hi
for each i “ 0, 1. Whenever txn u converges to x, the sequence of leaves tLpxn qu
converges to Lpxq in the Hausdorff distance. So hi is continuous.
For each g P L and x P R, we define ρpgqpxq P R by the formula:
Lpρpgqpxqq “ rpρ0 pgq ˝ h0 pxq, 0q, pρ1 pgq ˝ h1 pxq, 1qs .

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

95

Then it is routine to check that ρpgq P HomeoZ pRq for each g P L and ρ : L Ñ
HomeoZ pRq is a group homomorphism. Since
rpρ0 pgq ˝ h0 pxq, 0q, pρ1 pgq ˝ h1 pxq, 1qs
is the unique leaf containing pρpgqpxq, 1{2q, the relation ρ ěhi ρi follows. For the
backward direction, we simply use the map
hpxq “ h2 ˝ supph´1
1 pxqq.



Alternative proof of Lemma A.1 (2). One can give an elementary algebraic proof
based on the above idea. Define two strictly increasing maps
y ` hpy´q
y ` hpy`q
, f ` pyq “
.
2
2
Then f ´ and f ` are left- and right-continuous, respectively. Moreover, f ´ “ f `
except for the (countably many) jump discontinuities of h. So there uniquely exists
a continuous monotone map h0 which is the “inverse” of f ˘ in the following sense:
f ´ pyq “

f ´ ˝ h0 pxq ď x ď f ` ˝ h0 pxq, for all x.
Define
h1 pxq “ 2x ´ h0 pxq.
It is immediate that h0 and h1 are continuous and
hi ˝ T “ T ˝ hi .
If x ă y, then
h1 pxq “ 2x ´ h0 pxq ď hph0 pxq`q ď hph0 pyq´q ď 2y ´ h0 pyq “ h1 pyq.
So h1 is also monotone.
For each a P L, define
ρpaqpxq “

ρ0 paqh0 pxq ` ρ1 paqh1 pxq
.
2

We have seen that
h1 pxq P rhph0 pxq´q, hph0 pxq`qs.
Therefore we can write
ρpaqpxq P rp´ , p` s,
where
ρ0 paqh0 pxq ` ρ1 paq ˝ hph0 pxq˘q
2
ρ0 paqh0 pxq ` hpρ0 paq ˝ h0 pxq˘q
“
“ f ˘ pρ0 paqh0 pxqq.
2

p˘ “

96

S. KIM, T. KOBERDA, AND M. MJ

By the definition of h0 , we have
h0 ρpaqpxq “ ρ0 paqh0 pxq.
Moreover,
h1 ρpaqpxq “ 2ρpaqpxq ´ h0 ρpaqpxq “ 2ρpaqpxq ´ ρ0 h0 pxq “ ρ1 paqh1 pxq.
We see
ρ0 paqh0 ρpbqpxq ` ρ1 paqh1 ρpbqpxq
2
ρ0 paqρ0 pbqh0 pxq ` ρ1 paqρ1 pbqh1 pxq
“
“ ρpabqpxq.
2
Hence ρ is the desired group action.
ρpaqpρpbqxq “



Canonical Euler cocycle
The following lemma describes relations between the rotation (or translation) number and Euler (or canonical Euler) 2-cocycles.
Lemma A.2.

(1) For each g P Homeo` pS 1 q, we have
n
1ÿ
eupg, gk q
nÑ8 n
k“1

rot„ ˝spgq “ lim

in R,

n
1ÿ
rotpgq “ lim
eupg, gk q mod Z.
nÑ8 n
k“1

(2) As maps on L ˆ L, we have
τ “ eu ´ Bprot„ ˝sq.
Proof. (1) For g, h P Homeo` pS 1 q, observe that eupg, hq “ 1 if and only if
spgq ˝ sphqp0q ě 1.
Hence for each circle homeomorphism g, we have
n´1
n´1
ˇ
1 ÿ ˇˇ
1ÿ
k
k`1
ˇ
rot ˝spgq “ lim
pspgq p0q, spgq p0qs X Z “ lim
eupg, gk q.
n n
n n
k“0
k“0
„

The second statement obviously follows.
(2) For g, h P Homeo` pS 1 q, we have
τpg, hq “ rot„ pspgqsphqq ´ rot„ ˝spgq ´ rot„ ˝sphq
“ rot„ pspghqq ` eupg, hq ´ rot„ ˝spgq ´ rot„ ˝sphq
“ eupg, hq ` Bprot„ ˝sqpg, hq.



FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

97

Proof of Theorem 2.12, (4)ñ(5). There exists a bounded map β : L Ñ Z such that
ρ˚1 eu “ ρ˚0 eu ` Bβ.
By part (1) of Lemma A.2, for each g P L we have
rot„ ˝s ˝ ρ1 pgq “ lim
n

“ lim
n

n´1
1ÿ ˚
ρ eupg, gk q
n k“0 1
n´1
˘
1 ÿ` ˚
ρ0 eupg, gk q ` βpgq ` βpgk q ´ βpgk`1 q
n k“0

“ rot„ ˝s ˝ ρ0 pgq ` βpgq,
which proves the first equality. Moreover, we see from part (2) of the same Lemma
that
ρ˚1 τ “ ρ˚1 eu ´ Bprot„ ˝s ˝ ρ1 q “ ρ˚0 eu ` Bβ ´ Bprot„ ˝s ˝ ρ0 ` βq “ ρ˚0 τ.



Properties of a blow-up
Let us explain implications of having a common blow-up, closely following [24].
Let h : S 1 Ñ S 1 be a monotone degree one map. Following the notation from [24],
we let Gapphq denote the set of locally constant points of h, and
Corephq “ S 1 zGapphq.
Lemma A.3 ([24, Lemma 2.14]). If h is a non-constant monotone degree one map
on S 1 , then the set Corephq is perfect and uncountable.
Proof. This immediately follows from that Gapphq is a countable disjoint union of
open intervals such that no two intervals have a common endpoint.

Lemma A.4. Let ρ0 and ρ1 be circle actions of L.
(1) If ρ0 and ρ1 have a common blow-up, then we have that ρ0 ěh ρ1 for some
monotone degree one map h on S 1 , and that ρ˚0 eub “ ρ˚1 eub .
(2) If ρ0 and ρ1 are minimal, and if ρ0 ěh ρ1 for some monotone degree one
map h on S 1 , then ρ0 and ρ1 are conjugate.
Proof. (1) There exists a circle action ρ and surjective monotone degree one maps
h0 , h1 such that
ρ ěhi ρi .
We define for each x P S 1 ,
hpxq “ h1 ˝ inf ˝h´1
0 pxq.
The monotone degree one map h is well-defined since
∅ ‰ h0´1 pxq ‰ S 1 .

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S. KIM, T. KOBERDA, AND M. MJ

Claim. For each g P L and x P S 1 , we have
´1
ρpgq ˝ h´1
0 pxq “ h0 ˝ ρ0 pgqpxq.

Part Ď can be seen from
´1
h0 ˝ ρpgq ˝ h´1
0 pxq “ ρ0 pgq ˝ h0 ˝ h0 pxq “ ρ0 pgqpxq.

The opposite inclusion follows from
´1
´1
h0 ˝ ρpg´1 q ˝ h´1
0 ˝ ρ0 pgqpxq “ ρ0 pg q ˝ h0 ˝ h0 ˝ ρ0 pgqpxq “ x.

Now, we see ρ0 ěh ρ1 from the above claim and from
´1
ρ1 pgq ˝ hpxq “ ρ1 pgq ˝ h1 ˝ inf ˝h´1
0 pxq “ h1 ˝ ρpgq ˝ inf ˝h0 pxq “ h ˝ ρ0 pgqpxq.

For the second part, we may assume ρ0 is a blow-up of ρ1 . By Lemma A.3, we
may require that
0 P Corephq,
and furthermore, that 0 is not a boundary point of an interval in Gapphq.
Choose an arbitrary continuous monotone degree one map h̃ P HomeoZ pRq lifting h and let p “ h̃p0q. Let us denote the sections
sp f q “ f˜,

s p p f q “ fˆ.

so that
f˜p0q P r0, 1q,

fˆppq P rp, p ` 1q

for each f P Homeo` pS 1 q. Then the following diagram commutes modulo Z:
ρĆ
1 paq

/

R


h̃
ρz
1 paq

R

/

R


h̃

R.

We claim that the above diagram commutes on the nose. Evaluating at 0, we see
z
ρz
1 paqh̃p0q “ ρ1 paqp P rp, p`1q,

and that

p “ h̃p0q ď h̃ρĆ
0 paqp0q ď h̃p1q “ p`1.

Since h is not locally constant on either side of 0, we see
h̃ρĆ
0 paqp0q ‰ h̃p1q.
This establishes the commutativity of the diagram.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

99

For a, b P G, we have that
Ć
ρ˚0 eupa, bq “ 1 ô ρĆ
0 paqρ0 pbq0 ě 1
Ć
ô h̃ρĆ
0 paqρ0 pbq0 ě h̃1 “ p ` 1
z
ô ρz
1 paqρ1 pbqp ě p ` 1
ô ρ˚1 eu p pa, bq “ 1.
Note we used again the condition that 0 is not a locally constant point of h. It
follows that
ρ˚0 eu “ ρ˚1 eu p
as cocycles. By the basepoint independence of the bounded Euler class, we have
ρ˚0 eub “ ρ˚1 eub
in Hb2 pL; Zq.
(2) Let ρ0 ěh ρ1 for some monotone degree one map h on S 1 . Note that
`
˘
`
˘
ρ1 pGq hpS 1 q “ h ρ0 pGqpS 1 q “ hpS 1 q.
So hpS 1 q is ρ1 pGq–invariant. By minimality, we see hpS 1 q is dense in S 1 . It follows
that h does not have jump discontinuities, and so surjective and continuous. For
each open nonempty interval J in Gapphq, we see
h ˝ ρ0 pgqæ J “ ρ1 pgq ˝ hæ J
is constant. So Gapphq is ρ0 pGq–invariant. Since
Corephq “ S 1 zGapphq
is a closed, uncountable (Lemma A.3) and ρ0 pGq–invariant, we see that Corephq “
S 1 . So h is injective. This shows that h is a homeomorphism.

Finite orbits
Note that a circle action ρ of L has a finite orbit if and only if a finite-index subgroup
of L has a global fixed point. Recall the exponent of g in a group is the smallest
positive integer N such that gN “ 1.
Lemma A.5. Let ρ be a circle action of L.
(1) Then ρ has a finite orbit of cardinality N if and only if the exponent of ρ˚ eub
is N in Hb2 pL; Zq.
(2) If ρ has a finite orbit, then the map ρ̄ “ T ˝ rot ˝ρ is a group action of L and
satisfies ρ̄˚ eub “ ρ˚ eub .
Proof. (1) (ñ) Suppose
ρpGqp “ tp0 “ p, p1 , . . . , pN´1 u

100

S. KIM, T. KOBERDA, AND M. MJ

in the cyclic order of S 1 . Then we have a surjective group homomorphism β : G Ñ
Z{NZ such that ρpgqpi “ pi`βpgq . We may assume p “ 0. Fix a lift
β̃ : G Ñ t0, 1, . . . , N ´ 1u.
Then
ρ˚ eupa, bq “

˘
1
1`
β̃paq ` β̃pbq ´ β̃pabq “ B β̃pa, bq.
N
N

So we have Nρ˚ eub “ 0.
Let M be the exponent of ρ˚ eub “ 0. We have Mρ˚ eu “ B γ̃ for some bounded
map γ̃ : G Ñ Z. We can write N “ Mk for some k ą 0 by the minimality of M.
We see
ˆ
˙
`
˘
1
1
B β̃ ´ kγ̃ “ N
B β̃ ´ B γ̃ “ 0.
N
M
So β̃ ´ kγ̃ : G Ñ Z is a homomorphism, which is bounded. It follows that β̃ “ kγ̃.
Since β̃ is surjective, we have k “ 1 and M “ N.
(1) (ð) Suppose N is the exponent of ρ˚ eu. Note that we allow N “ 1. Then
Nρ˚ eu “ B β̃
for some bounded map
β̃ : G Ñ Z.
We have a group homomorphism
β : G Ñ Z{NZ
from the post-composition Z Ñ Z{NZ. The minimality of N implies that β is
surjective.
Set H “ ker β and σ “ ρæH . Note that
β̃paq{N P Z
for each a P H. By writing
σ˚ eupa, bq “ β̃paq{N ` β̃pbq{N ´ β̃pabq{N
we see that
σ˚ eub “ 0 P Hb2 pH; Zq.
Write the cocycle σ˚ eu “ Bγ for some bounded map γ on H. We can lift σ to
σ̃ : H Ñ HomeoZ pRq by the formula
σ̃pgq “ T p´γpgqqs ˝ σpgq;
see Section 2. Then
σ̃pgqp0q “ s ˝ σpgqp0q ´ γp0q P r´}γ}8 , }γ}8 ` 1q.
So we can find
p “ suptσ̃pgqp0q : g P Gu P Fix σ̃.

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

101

The projection of p is a global fixed point of σ. Moreover,
|ρpGqp| “ |G{H| “ N.
(2) Let ρpGqp be a finite orbit. Using the notations from the proof of (1), we
observe
rot ˝ρpgq “ βpgq{N.
So ρ̄ is a group homomorphism. While proving the direction (ñ) in (1), we have
already seen that
˘
1`
ρ˚ eu p pa, bq “
β̃paq ` β̃pbq ´ β̃pabq “ ρ̄˚ eu p pa, bq
N
as cocycles.

Lemma A.6. Let ρ0 , ρ1 be circle actions of L. Suppose either
(i) ρ0 and ρ1 have the same bounded Euler class, or
(ii) ρ0 and ρ1 have a common minimalization.
If ρ0 has a finite orbit, then so does ρ1 .
Proof. In the case (i), Lemma A.5 (1) implies the conclusion. In the case (ii), recall
that a minimalization of ρi is minimal if and only if ρ does not have a finite orbit. 
Proof of Lemma 2.11. Suppose ρ has an exceptional minimal set C; in particular, C
is a Cantor set. In this case, we let h P S 1 Ñ S 1 be the Cantor function which is
continuous and is locally constant on S 1 zC. Then we have ρ ěh ρ̄ for some minimal
action ρ̄. Uniqueness upfollows from Lemma A.4 implies that such ρ̄ is unique up
to conjugacy.
If ρ has a finite orbit, then Lemma A.5 implies that the minimalization ρ̄ is a
well-defined circle action.

For completeness, let us record a result that corresponds to Lemma A.6 in the
case of Homeo` pRq.
Lemma A.7. Let L be a countable group. If ρ0 , ρ1 P HompL, Homeo` pRqq are
semi-conjugate and if ρ0 has no global fixed point, then neither does ρ1 .
Proof. By Lemma A.1 (1), we may assume ρ0 ěh ρ1 for some surjective h. Let
us assume the contrary and pick y P Fix ρ1 . Then J “ h´1 pyq is a ρ0 pGq–invariant
proper connected subset of R. Since BJ consists of global fixed points of ρ0 , it
follows that J is empty. This contradicts the surjectivity of h.

Proof of Theorem 2.12. The implication (1)ñ(2) is immediate by Lemma A.1. The
proof of (4)ñ(5) was given right after Lemma A.2. If (1) holds, then Lemma 2.3
implies that (7) also holds.

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For (2)ñ(3), let ρ be a common blow-up of ρ0 and ρ1 , and let ρ̄i be a minimalization of ρi . We already saw that ρ˚0 eub “ ρ˚1 eub in Lemma A.4. By Lemma A.6,
we have two cases. If ρ0 and ρ1 have finite orbits, then
ρ̄i “ T ˝ rot ˝ρi .
Lemma 2.1 implies that ρ̄0 “ ρ̄1 . So we may assume neither have finite orbits and
ρ̄i is minimal. By Lemma A.4, we see ρ̄0 ěh ρ̄1 for some monotone degree one map
h, and moreover, ρ̄0 and ρ̄1 are conjugate.
For (3)ñ(4), suppose ρ0 and ρ1 have a common minimalization ρ̄. By Lemma A.6,
either both have finite orbits or ρ̄ is minimal. By applying Lemma A.5 (2) in the
first case and by applying Lemma A.4 (1) in the second, we see
ρ˚0 eub “ ρ̄˚ eub “ ρ˚1 eub .



For (2)ñ(6), we may assume ρ0 ěh ρ1 for some surjective monotone degree one
map h. As in [74, Proposition 7.6], we pick a path
tht : t P r0, 1qu Ď Homeo` pS 1 q
such that h0 “ Id and limtÑ1 ht “ h. Then tρt pgq “ ht ρ0 pgqh´1
t u is the desired path.
For (6)ñ (5), we note that the following is integer–valued:
rot„ ˝s ˝ ρt pgq ´ rot„ ˝s ˝ ρ0 pgq
By Lemma A.2, we see
ρ˚t τ ´ ρ˚0 τ “ ρ˚t eu ´ ρ˚0 eu ´ ps ˝ ρt q˚ B rot„ `ps ˝ ρ0 q˚ B rot„
is an integer–valued function on L ˆ L ˆ I. Since τ is continuous on Homeo` pS 1 q ˆ
Homeo` pS 1 q, we see that ρ˚t τ “ ρ˚0 τ, as desired.
The condition (5) implies (1), as elementarily (without using cohomology) proven
in [73, Theorem 5.11]. The implication (7)ñ(4) is included in [19, Theorem 1.4].
Acknowledgments
The authors thank M. Bestvina, B. Bowditch, D. Calegari, M. Casals-Ruiz, K.
Fujiwara, T. Ito, J. Kahn, M. Kapovich, I. Kim, Y. Matsuda, K. Ohshika, M.
Sakuma, Z. Sela, A. Sisto, M. Triestino, T. N. Venkataramana and H. Wilton for
helpful discussions. The authors thank M. Wolff for sharing [104] and for several
helpful comments. The authors are particularly grateful to K. Mann for a number
of insightful comments and for pointing out several references, particularly [34].
The second and third authors thank the Tata Institute for Fundamental Research in
Mumbai, and the first and second authors thank the Institute for Mathematical Sciences in Singapore, where parts of this research were conducted. The authors were
supported by the National Science Foundation under Grant No. DMS-1440140 at
the Mathematical Sciences Research Institute in Berkeley during Fall 2016, where

FLEXIBLITY OF GROUP ACTIONS ON THE CIRCLE

103

this research was completed. The first author is supported by Samsung Science and
Technology Foundation (SSTF-BA1301-06). The second author is partially supported by Simons Foundation Collaboration Grant number 429836, by an Alfred
P. Sloan Foundation Research Fellowship, and by NSF Grant DMS-1711488.. The
third author is supported in part by a Department of Science and Technology J.C.
Bose Fellowship grant.

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Department of Mathematical Sciences, Seoul National University, Seoul, Korea
E-mail address: s.kim@snu.ac.kr
URL: http://cayley.kr
Department of Mathematics, University of Virginia, Charlottesville, VA 22904-4137, USA
E-mail address: thomas.koberda@gmail.com
URL: http://faculty.virginia.edu/Koberda
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Colaba,
Mumbai 400005, India
E-mail address: mahan@math.tifr.res.in
URL: http://www.math.tifr.res.in/˜mahan/

