On laminar groups, Tits alternatives, and convergence group actions on S 2
OON LAMINAR GROUPS, TITS ALTERNATIVES,AND CONVERGENCE GROUP ACTIONS ON S JUAN ALONSO, HYUNGRYUL BAIK, AND ERIC SAMPERTONA
BSTRACT . Following previous work of the second author, we establish more properties of groups of circlehomeomorphisms which admit invariant laminations. In this paper, we focus on a certain type of such groups-so-called pseudo-fibered groups, and show that many 3-manifold groups are examples of pseudo-fibered groups.We then prove that torsion-free pseudo-fibered groups satisfy a Tits alternative. We conclude by proving that apurely hyperbolic pseudo-fibered group acts on the 2-sphere as a convergence group. This leads to an interestingquestion if there are examples of pseudo-fibered groups other than 3-manifold groups.
Keywords.
Tits alternative, laminations, circle homeomorphisms, Fuchsian groups, fibered 3-manifolds, pseudo-Anosov surface homeomorphism.
MSC classes:
1. I
NTRODUCTION
Thurston [24] showed that if M is an atoroidal 3-manifold admitting a taut foliation, then π ( M ) actsfaithfully on S with a pair of dense invariant laminations. The result was generalized by Calegari-Dunfield[6] and one can find a complete treatment in [5]. Motivated by these results, the second author studiedgroups acting faithfully on S with prescribed types and numbers of invariant laminations, in the processgiving a new characterization of Fuchsian groups [1]. In the same paper, he asked if there exists a way ofcharacterizing fibered 3-manifold groups in a similar way.For a 3-manifold M which fibers over the circle, one can construct a natural action of π ( M ) on thecircle wth two invariant laminations and this motivates the following definition. We call a subgroup G ofHomeo + ( S ) pseudo-fibered if its action on S admits two invariant, very full, loose laminations with distinctendpoints. As we said, this includes a large class of examples coming from 3-manifolds which fiber overthe circle, and the purpose of this paper is to show that pseudo-fibered groups in general have many niceproperties.We emphasize that in this paper all group actions on S are considered up to conjugacy not semi-conjugacy(compare [19]). This is because it is not clear if the notion of pseudo-fibering is invariant under semi-conjugacy. See Section 2 for the precise definitions and relevant discussions.In Section 4 we establish our first main result, a Tits alternative for torsion-free pseudo-fibered groups: Theorem A.
Let G be a torsion-free pseudo-fibered group. Each subgroup of G either contains a non-abelianfree subgroup or is virtually abelian.This is proved in Section 4.2 by studying how two elements of a (torsion-free) pseudo-fibered groupinteract with each other dynamically. In particular, we show a kind of dynamical alternative for elements ofa pseudo-fibered group: Theorem B.
Let G be a torsion-free pseudo-fibered group and for f ∈ G , let Per f denote the set of allperiodic points of f on S . Then for any g , h ∈ G , Per g and Per h are either equal or disjoint.Theorem A will follow from Theorem B by applying the ping-pong lemma, and H¨older’s theorem that agroup acting faithfully and freely on R is necessarily abelian.In Section 5, we study pseudo-fibered groups with more structure, inspired by (quasi-)Fuchsian groups.This part should be considered as part of Fenley’s program which generalizes the work of Cannon-Thurston[8] considerably from the viewpoint of pseudo-Anosov flows (see [12]). a r X i v : . [ m a t h . G T ] J a n JUAN ALONSO, HYUNGRYUL BAIK, AND ERIC SAMPERTON
Our main result of Section 5 connects the pseudo-fibered group action on S with a convegence groupaction on S . Note that a similar idea has been carried out in Fenley’s program (see [12], [13], and alsocompare [14], [23]). Theorem C.
Let G be a pseudo-fibered group which is purely hyperbolic. Then G acts on S as a conver-gence group.For the definition of “purely hyperbolic”, see Section 2. We expect that Theorem C can be strengthened.Indeed, when G is a purely hyperbolic pseudo-fibered group, the second author has previously conjecturedthat G is a Fuchsian group, hence acts on S as a convergence group [1]. Recall that the work of manyauthors (e.g. [25], [9] and [15]) shows that a group acts on S as a convergence group if and only if it istopologically conjugate to a Fuchsian group.Theorem A, Theorem B, and Theorem C show that pseudo-fibered groups have properties similar to thoseof fibered 3-manifold groups. On the other hand, we provide a source of examples of pseudo-fibered groupswhich are quite different from fibered 3-manifold groups in Section 3. More precisely, in Theorem 3.1 ,we show that the free product of any two finite cyclic groups is a pseudo-fibered group. In the context of3-manifolds, this implies that the fundamental group of the connected sum of any two lens spaces is alsopseudo-fibered (Corollary 3.4).Given the results above and the result of [1], we propose the following Conjecture 1.1 (Promotion of Pseudo-Fibering) . Let G be a finitely-generated torsion-free pseudo-fiberedgroup which does not split as a nontrivial free product. Then there are three possibilities. G is elementary, i.e. virtually abelian. G is topologically conjugate to a M¨obius group action (as usual, we consider
PSL ( R ) as a sub-group of Homeo + ( S ) ). G is abstractly isomorphic to a closed hyperbolic 3-manifold group.
A similar conjecture was made in [1] and the difference is discussed in Section 3. Here, G is said to beelementary if it is virtually abelian. This definition makes sense due to Theorem A, which asserts that everynon-elementary pseudo-fibered group contains a non-abelian free subgroup.1.1. Acknowledgements.
We thank David Cohen for asking the second author if the Tits alternative holdsfor pseudo-fibered groups. We also wish to thank Danny Calegari, Ursula Hamenst¨adt, and Dawid Kielak forhelpful conversations. We greatly appreciate for the anonymous referee for the valuable comments whichimproved the structure of the paper significantly. The second author was partially supported by the ERCGrant Nb. 10160104, and Samsung Science & Technology Foundation grant no. SSTF-BA1702-01.Thethird author thanks Universit¨at Bonn for hospitality, during which time some of this work was completed.2. P
RELIMINARIES
We briefly review and motivate several definitions regarding laminations on the circle. Two pairs ( a , b ) and ( c , d ) of distinct points of the circle S are said to be linked if each connected component of S \ { a , b } contains precisely one of c , d . They are called unlinked if they are not linked. Let M denote the set of allunordered pairs of two distinct points of S , i.e., M = ( S × S − ∆ ) / ( x , y ) ∼ ( y , x ) where ∆ is the diagonal { ( x , x ) : x ∈ S } . A lamination of S is a closed subset of M whose elements are pairwise unlinked. Givena lamination Λ , an element ( a , b ) of Λ is called a leaf , and the points a , b are called the endpoints of the leaf ( a , b ) (or just endpoints of Λ if there is no possible confusion). Two laminations have distinct endpoints iftheir sets of endpoints are disjoint. A lamination Λ is called dense if the set of endpoints of Λ is a densesubset of S .Any subgroup G of Homeo + ( S ) has an induced action on M . We say that a lamination Λ is G-invariant if the G -action on M preserves Λ set-wise. A discrete subgroup G of Homeo + ( S ) is called laminar if itadmits a dense G -invariant lamination. AMINAR GROUPS, TITS ALTERNATIVES, AND CONVERGENCE GROUP ACTIONS ON S Let D denote the closed unit disk in C where the interior is equipped with the Poincar´e metric, i.e., D = H ∪ ∂ ∞ H . A lamination Λ (cid:48) of D is a set of chords with disjoint interiors such that there exists alamination Λ in S = ∂ D where the chords in Λ (cid:48) can be obtained by connecting the endpoints of the leavesof Λ .As noted in Construction 2.4 of [5], the set of laminations on S and the set of geodesic laminationsof H are in one-to-one correspondence up to isotopy relative to S = ∂ ∞ H . Hence, we freely switchour viewpoint between these two without further mentioning. A gap of a lamination Λ is the closure of aconnected component of H \ Λ in H ∪ ∂ ∞ H .We recall some key properties of laminations from [1] (also compare [4]) . Definition 2.1.
A lamination Λ is said to be • totally disconnected if no open subset of the disk is foliated by Λ , • very full if each gap is a finite-sided ideal polygon in the disk, and • loose if no two leaves share an endpoint unless they are edges of the same (necessarily unique) gap. For every element f of Homeo + ( S ) , let Fix f ⊂ S denote the set of all fixed points of f . Let Per f denotethe set of all periodic points of f , where a point p of S is periodic for f if the orbit of p under f is finite.Thus Fix f ⊂ Per f . A fixed point p of the homeomorphism f is attracting if there exists an interval I (cid:51) p containing no other fixed points such that f ( I ) (cid:40) I . Similarly, a fixed point q is repelling if there exists aninterval J (cid:51) q containing no other fixed points such that f ( J ) (cid:41) J .We first give names to particular types of homeomorphisms of S in the following definition as in [1]. Forthe first four types of homeomorphisms, compare [18] where M¨obius-like elliptic, M¨obius-like parabolic,M¨obius-like hyperbolic, and M¨obius-like homeomorphisms are defined. Definition 2.2.
An element f of
Homeo + ( S ) is said to be • elliptic if f has no fixed points, • parabolic if f has a unique fixed point, • hyperbolic if f has two fixed points, one attracting and one repelling, • M¨obius-like if f is conjugate in
Homeo + ( S ) to an element of PSL ( R ) , • pseudo-Anosov-like or p-A-like if f is not hyperbolic and some positive power f n has a positive,even number of fixed points alternating between attracting and repelling, and • properly pseudo-Anosov-like or properly p-A-like if f is pseudo-Anosov-like and non-elliptic. Thus f is p-A-like if and only if a positive power of f is properly p-A-like. For a p-A-like homeomor-phism f ∈ Homeo + ( S ) , the set of boundary leaves of the convex hull of the attracting fixed points of aproperly p-A-like power f n is called the attracting polygon of f . Similarly, f has a repelling polygon . Definition 2.3.
Let Λ be a lamination. A leaf l ∈ Λ is said to be visible from a point p ∈ S if one canconnect l to p by a geodesic of H (i.e., there exists a geodesic ray from a point on l to p) which does notintersect any leaf of Λ in H . Observe that if p is an endpoint of a gap in a very full, loose lamination, the set of leaves visible from p is precisely the set of edges of the gap.Now we outline some examples of laminar groups mentioned in the introduction. Let S be a closedhyperbolic surface, and φ be a pseudo-Anosov homeomorphism of S . Let M be the mapping torus S × [ , ] / ( x , ) ∼ ( φ ( x ) , ) . Then one can construct a faithful action of π ( M ) on S in the following way. Note π ( M ) is isomorphic to π ( S ) (cid:111) Z . The deck transformation action of π ( S ) on H extends continuously toa faithful action on ∂ H . Let (cid:101) φ : H → H be a lift of φ to the universal cover of S . Since S has finite area, (cid:101) φ is a quasi-isometry, hence extends to a homeomorphism on ∂ H . Considering this homeomorphism on ∂ H as a generator of Z , this defines an action ρ : π ( M ) = π ( S ) (cid:111) Z → Homeo + ( ∂ H ) = Homeo + ( S ) . Then ρ ( π ( M )) is laminar, since it fixes both the stable and unstable laminations of φ . In fact, ρ is faithful. JUAN ALONSO, HYUNGRYUL BAIK, AND ERIC SAMPERTON
Definition 2.4.
A finitely generated laminar group G is said to be fibered if G is topologically conjugateto ρ ( π ( M )) where ρ and M are as in the previous paragraph, and the conjugacy takes the G-invariantlamination to one of the invariant laminations of the monodromy of M. Definition 2.5.
A finitely generated laminar group G is said to be pseudo-fibered if it preserves a pair ofvery full loose invariant laminations Λ , Λ with distinct endpoints, and each nontrivial element of G has atmost countably many fixed points in S . We also say ( G , Λ , Λ ) is a pseudo-fibered triple. Pseudo-fibered groups were first studied in [1], although they had not yet been given a name.
Theorem 2.6 (see Section 8 of [1]) . Let ( G , Λ , Λ ) be a pseudo-fibered triple. Let g ∈ G. Then (1) g is either M¨obius-like or pseudo-Anosov-like, and (2) if g is p-A-like, then for some i , j = , with i (cid:54) = j, Λ i contains the attracting polygon of g, and Λ j contains the repelling polygon of g. Furthermore, [1] shows that if G is torsion-free, then all M¨obius-like elements are hyperbolic elements.The following proposition justifies the term “pseudo-fibered”. Proposition 2.7.
A fibered group G is pseudo-fibered.Proof.
We only need to worry about the cardinality of the set of fixed points of each element. But this is not aproblem due to Theorem 5.5 of [10] which asserts that for a given pseudo-Anosov surface homeomorphism h , any lift of a strictly positive power of h has finitely many fixed points on ∂ ∞ H , alternating betweenattracting and repelling. (cid:3) In fact the proof of Theorem 5.5 in [10] (pp. 85-87) shows Theorem 2.6 in the case of fibered groups.Any lift of a strictly positive power of h falls into one of the three cases. Case 1 and Case 2 correspondto properly pseudo-Anosov-like elements and Case 3 corresponds to hyperbolic elements in the sense ofDefinition 2.2. In Case 1, the attracting repelling polygons of the p-A-like element have 3 or more sides andin Case 2, those polygons are degenerate, i.e., there are exactly two attracting fixed points and two repellingfixed points.We remark that the “pseudo” in “pseudo-fibered group” intentionally carries two different connotations.The first, as in Theorem 2.6, indicates that some elements are pseudo-Anosov-like. The second, as inConjecture 1.1, indicates that pseudo-fibered groups are (conjecturally) not far from fibered groups.In Section 5, we study a special class of pseudo-fibered groups. Definition 2.8.
Let G be a pseudo-fibered group. G is called purely hyperbolic if it has no pseudo-Anosov-like elements.
Theorem C says that a purely hyperbolic pseudo-fibered group acts on the sphere as a convergence group.In general, a group G acting on a compactum X is called a discrete convergence group if the following holds:for any infinite sequence of distinct elements ( g i ) of G , there exists a subsequence ( g i j ) of ( g i ) and two points a , b ∈ X not necessarily distinct such that g i j converges to the constant map with value a uniformly on everycompact subset of X \ { b } , and g − i j converges to the constant map with value b uniformly on every compactsubset of X \ { a } . Since we only deal with discrete convergence groups in this paper, we will omit the worddiscrete, and simply call it a convergence group.As mentioned before, when X is S , being a convergence group is equivalent to being (conjugate to) aFuchsian group. This result is known as the Convergence Group Theorem [25, 15, 9], and the same statementfor indiscrete convergence groups was proved in [17]. Remark 2.9.
There is a well-known equivalent definition of a convergence group action. G acting on X iscalled a convergence group if the diagonal action of G on X × X × X \ ∆ is properly discontinuous, where ∆ is the set of triples of points of X which are not all distinct. See, for instance, [26] . If the diagonal action onthe set of distinct triples is also cocompact, then G is called a uniform convergence group. AMINAR GROUPS, TITS ALTERNATIVES, AND CONVERGENCE GROUP ACTIONS ON S Remark 2.10.
When X is S , it is easy to see that the uniform convergence in the definition of a convergencegroup can be replaced by the pointwise convergence. However, it is important not to conflate the notions of“uniform convergence group” and “uniform convergence.” Remark 2.11.
When X is S , the analogue of Convergence Group Theorem is not true, i.e., not everydiscrete convergence group action on S comes from a Kleinian group. For instance, one can just start witha Fuchsian representation of a surface group into PSL ( C ) , and quotient the lower hemisphere to a singlepoint. On the other hand, it is a famous open problem whether or not all uniform convergence group actionson S come from Kleinian groups. Finally, the idea of a rainbow, first described in [1], will be useful throughout this paper:
Definition 2.12.
A lamination Λ is said to have a rainbow at a point p ∈ S if there is a sequence of leaves ( l i ) = (( a i , b i )) of Λ such that ( a i ) and ( b i ) converge to p from opposite sides. Such a sequence ( l i ) is calleda rainbow at p in Λ . A rainbow is a particularly nice way of approximating a point p ∈ S by leaves of a lamination. Clearlyendpoints of leaves do not admit rainbows. On the other hand, an observation we shall use later is that for avery full lamination Λ , these approximations exist for every point that is not an endpoint of a leaf: Lemma 2.13 (Rainbow Lemma, Theorem 5.3 of [1]) . Let Λ be a very full lamination of S . Every pointp ∈ S is either an endpoint of a leaf of Λ , or there is a rainbow in Λ at p. These two possibilities aremutually exclusive. (cid:3) Semi-conjugacy destroys pseudo-fiberedness.
We remark in this subsection that semi-conjugacy ap-pears to be irrelevant to the study of pseudo-fibered groups. This is to be expected in the context of ourpromotion of pseudo-fibering conjecture, since semi-conjugacy does not preserve convergence actions. Inparticular, our Theorem A should be seen as distinct from Margulis’s Tits alternative for minimal subgroupsof Homeo + ( S ) , a point we explain now.A continuous surjective map f : S → S is said to be monotone if the preimage of each point is connected.For a group G , two actions ρ and µ : G → Homeo + ( S ) are said to be semi-conjugate (or ρ is semi-conjugateto µ ) if there exists a monotone map f such that f ◦ ρ = µ ◦ f . For many aspects of the theory of groups ofcircle homeomorphisms, it is enough to consider the actions up to semi-conjugacy.A classical theorem of Poincar´e says that every subgroup of Homeo + ( S ) either has a finite orbit or issemiconjugate to a minimal action (meaning every orbit is dense). Furthermore, a theorem of Margulis saysthat subgroups of Homeo + ( S ) that act minimally either contain F as a subgroup or are abelian [20]. (Formore details regarding both of these results, as well as a general introduction to group actions on S , see[16].)What we have shown is that for a pseudo-fibered group, even if either there exists a finite orbit or theaction is non-minimal, the Tits alternative always holds. Hence, the scope of Theorem A is distinct from theTits alternative of Margulis.However, it is still an interesting question to ask if the pseudo-fibered groups can be studied up to semi-conjugacy. We observe that semi-conjugacy may destroy a pseudo-fibered triple, since the laminations donot behave well under semi-conjugacy. More precisely, we show: Proposition 2.14.
Let f : S → S be a monotone map which is not a homeomorphism, and ( Λ , Λ ) apair of laminations. At most one of the pairs, ( Λ , Λ ) and ( f ( Λ ) , f ( Λ )) , can be a pair of very full looselaminations with disjoint endpoint sets.Proof. Since f is not injective, there is a point p ∈ S such that I : = f − ( p ) has non-empty interior. Let ˆ p be an endpoint of I . Recall that the Rainbow Lemma says that for each p ∈ S and a very full lamination Λ ,either p is an endpoint of a leaf or there is a rainbow at p in Λ .Suppose ( Λ , Λ ) is a pair of very full loose laminations with disjoint endpoint sets. In particular, theremust be a rainbow at ˆ p in Λ i for at least one of the i = p under f is aninfinite set of leaves of f ( Λ i ) which share a common endpoint. Hence, f ( Λ i ) cannot be loose. JUAN ALONSO, HYUNGRYUL BAIK, AND ERIC SAMPERTON aaaa a bb F IGURE
1. The seedgraph Γ . . . . ... ... ... ... ... aaaa a bb ... ... ... ... ... ... . . . ... . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... abababab a ba ba b ba b a b baba baba ba b abb a b a F IGURE
2. The limiting circle S = Γ ∞ laminated by Λ , which is represented by the dashed red lines.For the other direction, suppose ( f ( Λ ) , f ( Λ )) is a pair of very full loose laminations with disjointendpoint sets. From the above argument, we know that there is no rainbow at ˆ p in Λ i for each i . But thismeans ˆ p is an endpoint of some leaf in both Λ and Λ , hence they cannot have disjoint endpoint sets. (cid:3) There are examples of pseudo-fibered triples whose actions are not minimal. Indeed, it is easy to constructexamples of pseudo-fibered groups with finite orbits, and in the next section, we construct examples ofpseudo-fibered groups whose actions are neither minimal nor have finite orbits. It would be interesting tomore thoroughly unravel the relationship between (non)minimal actions, finite orbits, and Conjecture 1.1.From this perspective, it is natural to ask that:
Question 2.15.
Is a pseudo-fibereing semi-conjugacy invariant? Namely, for two semi-conjugate actions ρ , ρ of a group G on S , if ρ is pseudo-fibered with two laminations Λ , Λ , is ρ also pseudo-fibered withrespect to a different pair of laminations Γ , Γ ? Note that even if the above question has an affirmative answer, the pairs ( Λ , Λ ) and ( Γ , Γ ) may not berelated in any obvious way as we saw in Proposition 2.14.3. F REE PRODUCTS , TORSION , AND PROMOTION OF PSEUDO - FIBERING
Theorems A and C can be seen also as partial evidence for Conjecture 1.1. In [1], a conjecture similar toConjecture 1.1 was made without free indecomposability assumption. The following theorem shows whyConjecture 1.1. was amended.
Theorem 3.1.
Let G , H be any finite cyclic groups. Then G ∗ H embeds into
Homeo + ( S ) as a pseudo-fibered group.Proof. This construction is adapted from a construction in [2] that yields a faithful action of any free productof subgroups of Homeo + ( S ) on a new circle, which blows down onto each of the original circles. Wecontent ourselves with a brief review of the ideas of [2], and a description of how to additionally constructinvariant laminations.To construct an action of G ∗ H on S , begin by forming two pointed copies of S called S G and S H , withmarked points both denoted 1. Let G act on S G as a finite rotation subgroup, and H act on S H as a finiterotation subgroup. The G -orbit of the point marked 1 in S G is now a copy of G , and the H -orbit of 1 in AMINAR GROUPS, TITS ALTERNATIVES, AND CONVERGENCE GROUP ACTIONS ON S ... ...... ... ... ... ... ... ...... ... ... ... ... ... ... ...... ... ... ... ... ... ... ... ...... ...... ...... ...... ...... ...... ...... ...... . . . ... ... ... ... ... ... ... ... ... ... ... . . . ... . . . ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ...... ...... ... ... ... ... ... ... ... ... ... ... ... ... ... ... ... F IGURE
3. The very full and loose lamination Λ ⊃ Λ . We have removed the markings toavoid clutter. S H is a copy of H . Mark all of these points accordingly, wedge S G and S H together at the points markedby 1, blow up all of the marked points, and consistenly label one of the endpoints of the blow-up intervals.The resulting “seed”, called Γ , is in Figure 1, where we have G = (cid:104) a | a = (cid:105) and H = (cid:104) b | b = (cid:105) . Nowgenerate an infinite graph Γ (cid:48) ∞ on which G ∗ H acts faithfully. As in Figure 2, write Γ (cid:48) ∞ = Λ ∪ Γ ∞ where Λ is the orbit of the blown-up intervals in Γ , and Γ ∞ is everything else. The order completion Γ ∞ is S , and Λ is a discrete lamination on this circle. This proves Lemma 3.2.
Let G , H be any finite cyclic groups. Then there exists an injective homomorphism ρ : G ∗ H → Homeo + ( S ) such that ρ ( G ∗ H ) admits a discrete invariant lamination. (cid:3) Now one can easily add more leaves to Λ to construct a G ∗ H -invariant very full and loose lamination Λ . For example, in the left circle of the seed Γ , first take a polygon which has one vertex in each connectedcomponent of the complement of the dotted segments and is invariant under the action of G . Then in theregion between this polygon and an element of Λ in Γ , add infinitely many triangles to make the laminationvery full and loose in that region. Now fill the other such regions so that lamination becomes G -invariant.One can do the same thing for the right circle to get an H -invariant very full loose lamination, and thenextend it as a G ∗ H -invariant lamination Λ which contains Λ as a sublamination. The final result is inFigure 3. Λ is obviously very full, and loose away from Λ . It is loose at Λ because the leaves of Λ arenot contained in gaps—they are instead limits of gaps.We need to construct another G ∗ H -invariant very full loose lamination Λ , so that Λ and Λ havedistinct endpoints. To build Λ , we first replace each leaf of Λ with endpoints by four leaves forming, say,a rectangle such that each endpoint of the original dotted segment lies between two adjacent vertices of therectangle. In the two regions between the rectangle and the endpoints of the original leaf in Λ , put infinitelymany triangles to make the lamination very full and loose. These choices can obviously be made so that JUAN ALONSO, HYUNGRYUL BAIK, AND ERIC SAMPERTON ... ... ...... F IGURE
4. Re-placing Λ withrectangles andtriangles. ... ...... ... ... ...... ... ... ... ...... ... ... ... ...... ...... ... ... ... ... ... F IGURE
5. The very full and loose lamination Λ .the endpoints of the new leaves are disjoint from Λ , and by working in one region at a time, we can do theconstruction G ∗ H -invariantly, resulting in Figure 4. In the regions where all the rectangles are visible, wedo the exactly same thing as when construction Γ from Γ : take a big invariant polygon, and fill out all thecomplementary regions. The result is shown in Figure 5.Finally, to show ( ρ ( G ∗ H ) , Λ , Λ ) is a pseudo-fibered triple, we need to show that every element of G ∗ H = ρ ( G ∗ H ) has countably many fixed points in its action on S . There are two ways to prove this,either using Bass-Serre theory, or the existence of even more G ∗ H -invariant laminations under this action.For the latter approach, we quote the following result: Theorem 3.3 ([1]) . Every subgroup of
Homeo + ( S ) admitting three very full invariant laminations withdistinct endpoint sets is M¨obius-like, meaning every element is (individually) M¨obius-like. (cid:3) Since both G and H are finite, there is a large freedom to construct laminations inductively as before.Indeed, we can slightly perturb the construction of Λ to get a third very full and loose invariant lamination Λ with endpoints distinct from Λ and Λ . Theorem 3.3 then implies every element of G ∗ H is M¨obius-like,hence has finitely many fixed points.Alternatively, to show every element of G ∗ H acts on S with at most two fixed points, we can use Bass-Serre theory. Clearly torsion elements of G ∗ H act freely on S . Nontorsion elements must have their fixedpoints in the subset S \ Γ ∞ , which can be identified with the ends of the Bass-Serre tree for G ∗ H . Standardresults now imply such elements have two fixed points in S . (cid:3) Theorem 3.1 has an immediate corollary in the context of 3-manifold groups.
Corollary 3.4.
Let M be a connected sum of two lens spaces. Then π ( M ) admits a pseudo-fibered groupaction on S .
4. T HE T ITS A LTERNATIVE FOR P SEUDO -F IBERED G ROUPS
Throughout this section, G will denote a torsion-free pseudo-fibered group, except where explicitly indi-cated elsewhere. This will allow us to apply Theorem 2.6, which we may do sometimes without mentioning.4.1. Proof of Theorem B.
To prove that pseudo-fibered groups satisfy the Tits alternative, we first proveTheorem B, which we recall says that two elements of a pseudo-fibered group have either equal or disjointsets of periodic points. This can be done, for instance, by analyzing how each element of the group acts onthe quotient of the circle obtained by collapsing leaves of an invariant lamination. One can show that sucha quotient is a dendrite as in [27], and this point of view has its own advantages. But for our purpose, it issimpler to analyze the group action on the circle directly.
AMINAR GROUPS, TITS ALTERNATIVES, AND CONVERGENCE GROUP ACTIONS ON S We will need a number of lemmas. The first follows immediately from the definitions, so we leave itsproof to the reader:
Lemma 4.1.
Suppose there are two dense laminations with distinct endpoints. Then each of the laminationsis totally disconnected. (cid:3)
Recall that by definition, a lamination Λ is a type of closed subset of ( S × S − ∆ ) / ( x , y ) ∼ ( y , x ) . Thus itmakes sense to talk about the neighborhood in Λ of a leaf, isolated leaves, etc . Lemma 4.2.
Each leaf of a totally disconnected very full lamination is either a boundary leaf of a gap, oris the limit of an infinite sequence of gaps.Proof.
This is a direct consequence of the definition of a very full lamination. Indeed, if a neighbourhoodof a leaf meets no gaps, it must be foliated, contradicting that the lamination is totally disconnected. (cid:3)
Lemma 4.3.
Let Λ be a totally disconnected very full lamination. Then Λ is loose if and only if the followingconditions are satisfied: (1) For each p ∈ S , at most finitely many leaves of Λ have p as an endpoint. (2) There are no isolated leaves.Proof.
Suppose Λ is loose. Then each p ∈ S is an endpoint of at most two leaves (maybe none) of Λ . Hencecondition (1) follows immediately.For condition (2), suppose there exists a leaf L which is isolated. Let J and J be the connected compo-nents of the complements of the endpoints of L in S . The fact that L is isolated means there exists an openarc I containing one endpoint of L , and another open arc I containing the other endpoint of L , such that foreach i = ,
2, there exists no leaf connecting I ∩ J i to I ∩ J i . For i = ,
2, define Λ i to be the set of leaves of Λ with endpoints in J i that are visible from both endpoints of L . Both Λ i are nonempty since Λ is dense. Foreach i = , Λ i ∪ { L } is the set of boundary leaves of a gap P i of Λ . This contradicts looseness of Λ , sinceboth P and P are gaps sharing some of their vertices.Now for the converse, assume Λ satisfies the conditions (1) and (2). Suppose p is a common endpoint oftwo gaps. Let L , L be the innermost leaves ending at p . (It’s possible L = L , but this does not changewhat follows.) Then L is either isolated (absurd by condition (2)), or is approximated by infinitely manyleaves. Since these leaves cannot cross L , they must end at p , contradicting condition (1). (cid:3) Lemma 4.4.
Let G be a pseudo-fibered group, and let Λ be a very full loose G-invariant lamination. If g isa hyperbolic element, then the fixed points of g in S are not an endpoint of any leaf of Λ .Proof. Let p , q be the fixed points of g . Suppose p is an endpoint of a leaf l .If the other endpoints of l is not q , then the set { g ◦ n ( l ) : n ∈ Z } gives an infinite set of leaves each ofwhich has p as an endpoint, contradicting Lemma 4.3. Hence, we may assume the other endpoint of l is q .Let I be a connected component of S \ { p , q } . Define Λ I to be the set of leaves whose endpoints are in I , and visible from both p and q . Assume Λ I is nonempty. Since { L } ∪ Λ I bound a gap, say P , Λ I mustbe finite. But this means there must be a leaf in Λ I which connects p to a point in I which we already sawimpossible. Now assume Λ I is empty. This means there exists a family of infinitely many leaves containedin I which accumulate to l . But since g acts as a translation on I , this is impossible (if l (cid:48) is a leaf closeenough to l , then l (cid:48) and gl (cid:48) must be linked). (cid:3) Theorem 4.5 (Solodov [22]) . If G is a subgroup of
Homeo + ( R ) such that each non-trivial element hasat most one fixed point, and there is no global fixed point, then [ G , G ] − { Id } consists of fixed-point-freeelements.Proof. See Step 4 in the proof of Theorem 2.2.36 in [22]. (cid:3)
Lemma 4.6.
Let ( G , Λ , Λ ) be a pseudo-fibered triple. Suppose g ∈ G is properly pseudo-Anosov-like. Ifh ∈ G shares a fixed point p with g, then
Fix h = Fix g . In particular, h is also a properly p-A-like element. Proof.
First, we show that h cannot be hyperbolic. Since p is a fixed point of g , then by Theorem 2.6 one of Λ or Λ contains a gap which has p as a vertex. But then Lemma 4.4 says that p cannot be a fixed point ofan hyperbolic element. Therefore, by Theorem 2.6, h must be pseudo-Anosov-like.Without loss of generality, we may assume that p is an attracting fixed point of both g , h , and Λ containsthe attracting polygon P g of g . Let q be a vertex of P g which is connected to p by a boundary leaf l of P g .If q is not a fixed point of h , then { h ◦ n ( l ) : n ∈ Z } is an infinite set of leaves which share p as a commonendpoint. This is impossible by Lemma 4.3 since Λ is loose. This inductively shows that all vertices (i.e.,all attracting fixed points of g ) are fixed by h . Theorem 2.6 says there is no leaf connecting an attractingfixed point to a repelling fixed point for a given p-A-like elements. Hence, all attracting fixed points ofelements of g are attracting fixed points of h . Applying the same argument to the attracting polygon of h ,one concludes that P g is in fact the attracting polygon of h as well.We showed the attracting polygon of g and the attracting polygon of h must coincide if p is attracting.How about the repelling polygons? Note Theorem 2.6(2) implies both the repelling polygon of g and therepelling polygon of h are contained in Λ .Case 1. Suppose g (hence, also h ) has at least three attracting fixed points. Since both g and h has aunique repelling fixed point between two adjacent vertices of P g , the only way for the repelling polygons tobe unlinked is that they coincide. Therefore, Fix g = Fix h .Case 2. Now suppose g has only two attracting fixed points. We know that each connected componentsof S \ Fix g has exactly one repelling fixed point of g and exactly one repelling fixed point of h . Let I be aconnected component of S \ Fix g , r g the repelling fixed point of g on I , r h the repelling fixed point of h on I . Let H be the subgroup of G generated by g , h . First note that every element of H fixes the endpoints of I . From the previous arguments, we know that each element of H is p-A-like and has exactly two attractingfixed points and two repelling fixed points. This implies that each element of H has exactly one fixed pointin I . Now we apply Theorem 4.5 by identifying I with R . If r g (cid:54) = r h , H has no global fixed point in I , and thetheorem says that H is abelian. But if g and h commute, then r g and r h coincide, a contradiction. Therefore, r g and r h must coincide from the beginning, and Fix g = Fix h again. (cid:3) The following generalizes Lemma 4.6 for p-A-like elements that could be elliptic. We use Per f to denotethe set of all periodic points of a homeomorphism f . Lemma 4.7.
Let g be a p-A-like element in a pseudo fibered group G. If h ∈ G shares a periodic point pwith g, then
Per h = Per g and h is also p-A-like.Proof. Take powers g n and h m such that g n is properly p-A-like and p is fixed for h m . Note that everyperiodic point of a properly p-A-like element is fixed. So Fix g n = Per g n = Per g . Applying Lemma 4.6, weget that Fix g n = Fix h m and that h m is properly p-A-like. So h is p-A-like and we also have Per h = Fix h m , thatagrees with Fix g m = Per g . (cid:3) The previous lemma establishes “half” of the dynamical alternative. The other half follows from the nextlemma.
Lemma 4.8.
Let G be a pseudo-fibered group. Suppose g , h are hyperbolic elements of G which share afixed point p. Then every element of the subgroup generated by g , h has the same fixed points as g.Proof. Since p is fixed by g , h , g − , h − , any element of the subgroup H of G generated by g , h fixes p . ButLemma 4.6 says no p-A-like element shares a fixed point of a hyperbolic element. Hence, all elements ofsuch a subgroup must be hyperbolic.Now we apply Theorem 4.5 to H by identifying S \ { p } with R . Just as in Case 2 of the proof of Lemma4.6, we conclude that H has a (unique) global fixed point in S \ { p } . (cid:3) Combining Lemmas 4.7 and 4.8 with Theorem 2.6, we immediately conclude Theorem B. (cid:3)
AMINAR GROUPS, TITS ALTERNATIVES, AND CONVERGENCE GROUP ACTIONS ON S Proof of Theorem A.
We now combine Theorem B with two known results to prove Theorem A. Thefirst result is a very well-known tool in geometric group theory (for instance, see Ch. II.B of [11]):
Theorem 4.9 (Ping-pong lemma) . Let G be a group acting on a set X . Let g , g be elements of G. Supposethere exist disjoint nonempty subsets X + , X − , X + , X − of X such that g i ( X − X − i ) ⊂ X + i , g − i ( X − X + i ) ⊂ X − i for each i = , . Then the subgroup generated by g , g is free. (cid:3) A proof of the second result we need can be found in many places, e.g. [16] or [22]:
Theorem 4.10 (H¨older) . Let K be a subgroup of
Homeo + ( R ) which acts freely on R . Then K is abelian. (cid:3) Now the ping-pong lemma implies the first case of the Tits alternative:
Lemma 4.11.
Let G be a pseudo-fibered group and g , g ∈ G. If
Per g and Per g are disjoint, then thereare powers of g and g that generate a non-abelian free subgroup of G.Proof. We can replace g and g by some powers that satisfy Per g i = Fix g i . (If g i is hyperbolic, or properly p-A-like, no power needs to be taken. If g i is elliptic p-A-like, take a power that is properly p-A-like). Take X + i to be a neighborhood of the attracting fixed points of g i and X − i to be a neighborhood of the repelling fixedpoints of g i (for i = , g and Fix g are disjoint by hypothesis, we can take X + i , X − i (for i = , h i be a high enough power of g i so that h i ( S − X − i ) ⊂ X + i and h − i ( X − X + i ) ⊂ X − i ,for i = ,
2. This is possible because of the dynamics of hyperbolic and properly p-A-like elements. By thePing-Pong lemma, h and h generate a free subgroup. (cid:3) H¨older’s theorem implies the alternative case:
Lemma 4.12.
Let G be a pseudo-fibered group, g ∈ G and P = Per g . Consider the subgroup H = { h ∈ G : hP = P } of G that leaves P invariant. Then: (1) H = { h ∈ G : Per h = P } (2) H is virtually abelian.Proof.
Let h ∈ H . Since hP = P and P = Per g is finite, then P ⊂ Per h . Then by Theorem B we get thatPer h = Per g = P . This proves the first assertion, the other inclusion being trivial.For the second assertion, let K (cid:69) H consist of elements that stabilize Per g pointwise. That is, K = { h ∈ H | Fix h = Per g } . Note [ H : K ] ≤ | Per g | , so in particular, to show H is virtually abelian, it suffices to show K is abelian. Assertion (1) implies K acts freely on each component of S \ Per g , each of which is an interval. So by H¨older’s theorem, K isabelian. (cid:3) Theorem A now follows from Theorem B, since if H is a subgroup of G , then either there exist twoelements of H with distinct periodic point sets, or H has a global set of periodic points. In the first case,apply Lemma 4.11; in the second case, apply Lemma 4.12. (cid:3) Remarks on the proofs of Theorems A and B.
Conjecturally, non-elementary pseudo-fibered groupsare word-hyperbolic. For word-hyperbolic groups, a stronger version of the Tits alternative holds: an infinitesubgroup of a word-hyperbolic group either contains a free group of rank 2 or is virtually cyclic . To obtainthis stronger Tits alternative, one needs to strengthen Lemma 4.12. Let H be a subgroup as in Lemma4.12, and assume it is actually abelian. Then Ghys [16] provides an H -invariant measure on each connectedcomponent of S − Per H . The problem is that this measure may not have full support. Even when the actionof G on S is minimal, it is still not clear if it can be shown that we have an invariant measure of full supporton the complement of Per H . The stronger Tits alternative would easily follow from there. More directly, it is easy to verify that this stronger Tits alternative is equivalent to showing that thesubgroup K in the proof of Lemma 4.12 is isomorphic to Z . A stronger version of H¨older’s theorem saysthat for any abelian subgroup of Homeo + ( R ) , there is a blowdown of R such that the induced action of thesubgroup is faithful and by translations. Thus, showing K is Z is equivalent to showing that this translationaction is not minimal.Finally, we remark that if we knew that pseudo-Anosov-like elements really were pseudo-Anosov, thestronger Tits alternative would follow from the fact that two pseudo-Anosovs commute if and only if theyare powers of some other pseudo-Anosov. This can be seen by considering the action of the mapping classgroup on Thurston’s compactification of Teichm¨uller space: if two pseudo-Anosovs commute, they must fixthe same axis. The mapping class group acts discretely on Teichm¨uller space, hence, the subgroup generatedby the two pseudo-Anosovs must act discretely on the axis in Teichm¨uller space. Each pseudo-Anosov actsby translations on this axis, so we conclude that the subgroup the two generate is isomorphic to Z .5. P URELY HYPERBOLIC P SEUDO - FIBERED GROUPS AND CONVERGENCE GROUP ACTIONS ON S We now restrict our attention to the special class of purely hyperbolic pseudo-fibered groups. In [1], it wasconjectured that such groups are always Fuchsian, or, equivalently, convergence subgroups of Homeo + ( S ) .While this conjecture remains open, Theorem C shows that, as expected, purely hyperbolic pseudo-fiberedgroups act on the 2-sphere as convergence groups. We remark that a similar idea of relating group actionon S with a geometric origin to convergence group action on S has been carried out in the context ofpseudo-Anosov flows in Fenley’s program (see Section 4 of [12]).To prove Theorem C, we begin by reviewing the construction of [1], which is responsible for the existenceof an S on which a pseudo-fibered group can act, and inspired by results of Cannon and Thurston [8].Moore’s theorem [21] implies that for any pseudo-fibered triple ( G , Λ , Λ ) , there exists a quotient map π : S → S , constructed by first identifying two disks laminated by Λ and Λ (respectively) along theircommon boundary S , and then collapsing all the gaps of the Λ i to points. Since each lamination is G -invariant, this induces a G -action on S such that π is G -equivariant. We call this map π the Cannon-Thurston map for the pseudofibered triple ( G , Λ , Λ ) . For details, one can also consult Section 14 of [8]. Abasic observation about this construction is Lemma 5.1.
Let ( G , Λ , Λ ) be a pseudo-fibered triple. Suppose there exists a sequence ( x i ) of points inS (= π ( S )) which converges to x, and a sequence ( g i ) of elements of G such that g i ( x i ) converges to x (cid:48) inS . Then, passing to subsequences if necessary, there exists a sequence ( x i ) of points in S converging to xsuch that g i ( x i ) converges to x (cid:48) in S , where x i = π ( x i ) and x (cid:48) = π ( x (cid:48) ) .Proof. This is straightforward because S is compact, and π is continuous, surjective and G -equivariant. (cid:3) We now state a few dynamical lemmas, after which we will prove Theorem C.
Lemma 5.2.
Let G be a group acting on S such that there exists a G-invariant lamination with a rainbowat p ∈ S . Suppose there exists a sequence ( g i ) of elements of G such that for any neighborhood U of p,g i ( U ) intersects U nontrivially for all large i. Then p is an accumulation point of some fixed points of theelements in the sequence ( g i ) .Proof. See the proof of Proposition 7.5 in [1]. (cid:3)
Lemma 5.3.
Let ( G , Λ , Λ ) be a pseudo-fibered triple. Suppose ( x i ) is a sequence of points in S whichconverges to x, and there exists a sequence ( g i ) of elements of G such that g i ( x i ) converges to x (cid:48) . Then eitherx is an accumulation points of fixed points of the sequence ( g − i + ◦ g i ) or x (cid:48) is an accumulation point of fixedpoints of the sequence ( g i + ◦ g − i ) . Moreover, if x i = x for all i, then x (cid:48) must be an accumulation point offixed points of (any subsequence of) the sequence ( g i + ◦ g − i ) .Proof. This is a straightforward consequence of Lemma 5.2. See, for instance, the proof of Proposition 7.6in [1]. (cid:3)
AMINAR GROUPS, TITS ALTERNATIVES, AND CONVERGENCE GROUP ACTIONS ON S Proof of Theorem C.
Suppose the negation of the conclusion. Then there exists an infinite sequence ofdistinct elements ( g i ) of G which does not have the convergence property, i.e. , the set { g i } does not actproperly discontinuously on the set of triples of distinct points of S . More precisely, this means that, afterpassing to a subsequence of ( g i ) , there exist three convergent sequences in S x i → x , y i → y , z i → z , x i (cid:54) = y i (cid:54) = z i (cid:54) = x i , x (cid:54) = y (cid:54) = z (cid:54) = x , and three elements x (cid:48) , y (cid:48) , z (cid:48) ∈ S such that g i ( x i ) → x (cid:48) , g i ( y i ) → y (cid:48) , g i ( z i ) → z (cid:48) , x (cid:48) (cid:54) = y (cid:48) (cid:54) = z (cid:48) (cid:54) = x (cid:48) . We conclude that there exist sequences and points in S such that x i → x , y i → y , z i → z , x i (cid:54) = y i (cid:54) = z i (cid:54) = x i , x (cid:54) = y (cid:54) = z (cid:54) = x and three elements x (cid:48) , y (cid:48) , z (cid:48) ∈ S such that g i ( x i ) → x (cid:48) , g i ( y i ) → y (cid:48) , g i ( z i ) → z (cid:48) , x (cid:48) (cid:54) = y (cid:48) (cid:54) = z (cid:48) (cid:54) = x (cid:48) , where in our notation, p is some fixed point in the preimage of p under π for any p ∈ S , in accordance withLemma 5.1. In words, we can lift sequences exhibiting the failure of G to act as a convergence group on S to sequences exhibiting the failure of G to act as a convergence group on S .Since Λ and Λ do not share any endpoints, for each p ∈ { x , y , z } , there exists a rainbow at p in at leastone of the Λ i . In particular, for each p ∈ { x , y , z } there exists a leaf L p which separates p from the other twopoints in { x , y , z } \ { p } (which lamination L p belongs to is not important, and L x , L y , L z are not necessarilyleaves of the same lamination). Passing to a subsequence, we may assume that each of the sequences ofpairs of points described by the endpoints of the leaves ( g i ( L x )) , ( g i ( L y )) , ( g i ( L z )) , ( g − i ( L x )) , ( g − i ( L y )) , ( g − i ( L z )) converges to a pair of points, which are possibly not distinct.By Lemma 5.3, either at least two of x , y , and z are accumulation points of fixed points of the sequence ( g − i + ◦ g i ) or at least two of x (cid:48) , y (cid:48) , and z (cid:48) are accumulation points of fixed points of the sequence ( g i + ◦ g − i ) .Without loss of generality, we assume that x (cid:48) and y (cid:48) are accumulation points of fixed points of the sequence ( g i + ◦ g − i ) , possibly after exchanging the roles of g i and g − i . Furthermore, since each element of G hasexactly two fixed points, there exists a subsequence ( g i j + ◦ g − i j ) of the sequence ( g i + ◦ g − i ) such that x (cid:48) and y (cid:48) are the only accumulation points of the fixed points of the g i j + ◦ g − i j . By the second statement of Lemma5.3, if p ∈ S is such that g i ( p ) converges to p (cid:48) , then p (cid:48) must be either x (cid:48) or y (cid:48) . In particular, consideringthe sequence ( g i ( L z )) , which converges to some pair of possibly nondistinct points { e , e } , what we haveshown implies each e i is either x (cid:48) or y (cid:48) .If e (cid:54) = e , since laminations are required to be closed, the limit of the sequence of leaves ( g i ( L z )) ) must bea leaf connecting x (cid:48) to y (cid:48) in the lamination Λ i containing L z . But since x (cid:48) = π ( x (cid:48) ) and y (cid:48) = π ( y (cid:48) ) are assumedto be distinct, by the definition of the Cannon-Thurston map π , their preimages cannot be connected by aleaf. This contradiction implies e = e .Let’s assume that e = e = x (cid:48) . We will show that x (cid:48) is not distinct from both y (cid:48) and z (cid:48) . In the case e = e = y (cid:48) , the same argument would lead us to contradict that y (cid:48) is not distinct from both x (cid:48) and z (cid:48) .Let I y be the closure of the connected component of S − L z which contains y , and define I z similarlyfor z . Take a nested sequence of closed neighborhoods ( U i ) of x (cid:48) such that U i → x (cid:48) as i → ∞ . Passing to asubsequence, one may assume that the endpoints of g i ( L z ) are contained in U i .For each i , there are two possibilities: either g i ( I z ) ⊂ U i , or g i ( I y ) ⊂ U i . Suppose the former happens forinfinitely many i . Since ( z i ) converges to z , then for all large enough i , z i is in I z . Hence, g i ( z i ) ∈ U i forinfinitely many i , so some subsequence of g i ( z i ) converges to x (cid:48) . But this is impossible since z (cid:48) is assumedto be distinct from x (cid:48) , and g i ( z i ) converges to z (cid:48) . If instead g i ( I y ) ⊂ U i for infinitely many i , then we similarlycontradict the assumption that y (cid:48) (cid:54) = x (cid:48) . (cid:3) We remark that this proof almost goes through to show that a purely hyperbolic pseudo-fibered group G acts as a convergence group on the circle S = π − ( S ) . The only gap to consider is the case where e (cid:54) = e .We conclude the paper with some questions which naturally arise from the results of this paper. Question 5.4.
For fibered groups, how do hyperbolic elements and pseudo-Anosov-like elements interact?What can be say about the group structure of a pseudo-fibered group in terms of the dynamical feature ofthe elements of the group?
Question 5.5.
For a fibered group action on S , without using hyperbolic geometry, can we abstractly showthat the induced action on S as in this section is a uniform convergence group? Answering (or rather understanding) above questions appropriately, one might hope to get a characteri-zation of fibered hyperbolic 3-manifold groups via its action on S with invariant laminations. For instance,assuming Cannon’s conjecture [7], an affirmative answer to Question 5.5 together with a result of Bowditch[3] would imply that such a group is always a closed hyperbolic 3-manifold group.R EFERENCES [1] H. Baik. Fuchsian groups, circularly ordered groups, and dense invariant laminations on the circle.
Geom. Topol. , 19(4):2081–2115, 2015.[2] H. Baik and E. Sampterton. Spaces of invariant circular orders of groups. to appear in Groups Geom. Dyn.[3] B. Bowditch. A topological characterization of hyperbolic groups.
J. Amer. Math. Soc. , 11:643–667, 1998.[4] D. Calegari. Foliations and geometrization of 3-manifolds. Lecture note for the course ’Foliations and 3-manifolds’ at Uni-versity of Chicago, 2003.[5] D. Calegari.
Foliations and the Geometry of 3-Manifolds . Oxford Science Publications, 2007.[6] D. Calegari and N. Dunfield. Laminations and groups of homeomorphisms of the circle.
Invent. Math. , 152:149–207, 2003.[7] J. W. Cannon and E. L. Swenson. Recognizing constant curvature discrete groups in dimension 3.
Trans. Amer. Math. Soc. ,350(2):809–849, 1998.[8] J. W. Cannon and W. P. Thurston. Group invariant Peano curves.
Geometry & Topology , 11:1315–1355, 2007.[9] A. Casson and D. Jungreis. Convergence groups and Seifert fibered 3-manifolds.
Invent. math. , 118:441–456, 1994.[10] A. J. Casson and S. A. Bleiler.
Automorphisms of surfaces after Nielsen and Thurston , volume 9. Cambridge University Press,1988.[11] P. de la Harpe.
Topics in geometric group theory . Chicago Lectures in Mathematics. University of Chicago Press, 2000.[12] S. Fenley. Ideal boundaries of pseudo-anosov flows and uniform convergence groups, with connections and applications tolarge scale geometry.
Geom. Topol. , 17:1–110, 2012.[13] S. Fenley. Quasigeodesic pseudo-anosov flows in hyperbolic 3-manifolds and connections with large scale geometry.
Advancesin Mathematics , 303:192–278, 2016.[14] S. Frankel. Quasigeodesic flows and sphere filling curves. arXiv:1210.7050.[15] D. Gabai. Convergence groups are Fuchsian groups.
Bull. Amer. Math. Soc. , 25(2):395–402, 1991.[16] E. Ghys. Groups acting on the circle.
L’Enseignement Math´ematique , 47:329–407, 2001.[17] A. Hinkkanen. Abelian and nondiscrete convergence groups on the circle.
Trans. Amer. Math. Soc. , 318:87–121, 1990.[18] N. Kovaˇcevi´c. Examples of m¨obius-like groups which are not m¨obius groups.
Trans. Amer. Math. Soc. , 351(12):4823–4835,1999.[19] K. Mann. Spaces of surface group representations.
Invent. Math. , 201(2):669–710, 2015.[20] G. Margulis. Free subgroups of the homeomorphism group of the circle.
C. R. Acad. Sci. Paris S´er. I Math. , 331(9):669–674,2000.[21] R. L. Moore. Concerning upper semi-continuous collections of continua.
Trans. Amer. Math. Soc. , 27:416–428, 1925.[22] A. Navas.
Groups of Circle Diffeomorphisms . Chicago Lectures in Mathematics. The University of Chicago Press, 2011.[23] S. F. Thierry Barbot. Free seifert pieces of pseudo-anosov flows. arXiv:1512.06341.[24] W. P. Thurston. Three-manifolds, Foliations and Circles, I. arXiv:math/9712268v1 [math.GT] , 1997.[25] P. Tukia. Homeomorphic conjugates of Fuchsian groups.
J. reine angew. Math. , 398:1–54, 1988.[26] P. Tukia. Convergence groups and Gromov’s metric hyperbolic spaces.
New Zealnd J. Math. , 23(2):157–187, 1994.[27] E. T. Winkel. Sticky and slippery laminations on the circle. Master’s thesis, University of Bonn, 2016.
AMINAR GROUPS, TITS ALTERNATIVES, AND CONVERGENCE GROUP ACTIONS ON S C ENTRO DE M ATEM ´ ATICA , F
ACULTAD DE C IENCIAS , U
NIVERSIDAD DE LA R EP ´ UBLICA . I GU ´ A ONTEVIDEO
C.P.11400, U
RUGUAY . E-mail address : [email protected] D EPARTMENT OF M ATHEMATICAL S CIENCES , KAIST, 291 D
AEHAK - RO Y USEONG - GU , D AEJEON , 34141, S
OUTH K O - REA
E-mail address : [email protected] D EPARTMENT OF M ATHEMATICS ,, S
OUTH H ALL , R
OOM
NIVERSITY OF C ALIFORNIA , S
ANTA B ARBARA , CA93106-3080
E-mail address ::