ENDING LAMINATIONS AND CANNONTHURSTON MAPS MAHAN MJ WITH AN APPENDIX BY SHUBHABRATA DAS AND MAHAN MJ Abstract
131K - views

ENDING LAMINATIONS AND CANNONTHURSTON MAPS MAHAN MJ WITH AN APPENDIX BY SHUBHABRATA DAS AND MAHAN MJ Abstract

In earlier work we had shown that CannonThurston maps exist for Kleinian surface groups without accidental parabolics In this p aper we prove that preimages of points are precisely endpoints of leaves of t he ending lamination whenever the CannonThu

Tags : earlier work
Download Pdf

ENDING LAMINATIONS AND CANNONTHURSTON MAPS MAHAN MJ WITH AN APPENDIX BY SHUBHABRATA DAS AND MAHAN MJ Abstract




Download Pdf - The PPT/PDF document "ENDING LAMINATIONS AND CANNONTHURSTON MA..." is the property of its rightful owner. Permission is granted to download and print the materials on this web site for personal, non-commercial use only, and to display it on your personal computer provided you do not modify the materials and that you retain all copyright notices contained in the materials. By downloading content from our website, you accept the terms of this agreement.



Presentation on theme: "ENDING LAMINATIONS AND CANNONTHURSTON MAPS MAHAN MJ WITH AN APPENDIX BY SHUBHABRATA DAS AND MAHAN MJ Abstract"— Presentation transcript:


Page 1
ENDING LAMINATIONS AND CANNON-THURSTON MAPS MAHAN MJ WITH AN APPENDIX BY SHUBHABRATA DAS AND MAHAN MJ Abstract. In earlier work, we had shown that Cannon-Thurston maps exist for Kleinian surface groups without accidental parabolics. In this p aper we prove that pre-images of points are precisely end-points of leaves of t he ending lamination whenever the Cannon-Thurston map is not one-to-one. Contents 1. Introduction 1.1. Statement of Results 1.2. Outline and Applications 2. Preliminaries 2.1. Hyperbolic Metric Spaces 2.2. Split Geometry 3. Laminations 10 3.1. Ideal points are

identified by Cannon-Thurston Maps 10 3.2. Leaves of Laminations 12 4. Closed Surfaces 13 4.1. Geodesic Laminations and -trees 13 4.2. Rays Contained in Ladders 14 4.3. Main Theorem for Simply Degenerate Groups 16 4.4. Modifications for Totally Degenerate Groups 17 4.5. Application: Rigidity 18 Appendix A (by Shubhabrata Das and Mahan Mj) Surfaces with Cus ps 18 References 21 1. Introduction 1.1. Statement of Results. In earlier work we showed: Theorem 1.1. [Mj14] Let PSL be a discrete faithful representa- tion of a surface group with or without punctures, and withou t accidental

parabolics. Let / )) . Let be an embedding of in that induces a homotopy equivalence. Then the embedding extends continuously to a map . Further, the limit set of )) is locally connected. 2010 Mathematics Subject Classification. 57M50, 20F67 (Primary); 20F65, 22E40 (Secondary).
Page 2
2 MAHAN MJ WITH SHUBHABRATA DAS This generalizes the first part of the next theorem due to Cann on and Thurston [CT85] (for 3 manifolds fibering over the circle) and Minsky [ Min94] (for bounded geometry closed surface Kleinian groups): Theorem 1.2. [CT85, CT07, Min94] Suppose a closed

surface group of bounded geometry acts freely and properly discontinuously on by hyperbolic isometries. Then the inclusion extends continuously to the boundary. Further, pre-images of points on the boundary are precisely ideal boundary points of a leaf of the ending lamination, or ideal boundary points of a complementary ideal polygon whenever the Cannon-Thurston map is not one-to-one In the main body of this paper, we generalize the second part o f the above theorem to arbitrary Kleinian closed surface groups withou t accidental parabolics. Theorem 1.3. Suppose a closed surface group acts

freely and properly dis- continuously on by hyperbolic isometries. Then the inclusion extends continuously to the boundary. Further, pre-images of point s on the boundary are precisely ideal boundary points of a leaf of the ending lamin ation, or ideal boundary points of a complementary ideal polygon whenever the Cannon -Thurston map is not one-to-one. In passing from Theorem 1.2 to Theorem 1.3 we have removed the hypothesis of bounded geometry . In an appendix to the paper we extend Theorem 1.3 to the case of surfaces with cusps (Theorem A.4). 1.2. Outline and Applications. We first

outline the main steps involved in the proof of the main Theorem 1.3. To fix notions, we let be the convex core of a simply or doubly degenerate hyperbolic 3-manifold homoto py equivalent to a surface . We also assume that an inclusion inducing the homotopy equivalence is fixed. Let denote the lift of to universal covers. Recapitulation of Theorem 1.1 from [Mj14] To show that a Cannon-Thurston map exists we have to show that extends con- tinuously to the boundary giving . The proof of the main Theorem 1.1 of [Mj14] proceeds (cf. Lemma 2.2 below) by showing that give n a geodesic

segment in (the intrinsic metric on) lying outside a large ball about a fixed reference point in , the hyperbolic geodesic in joining its end-points lies outside a large ball about ) in . Towards this a hyperbolic ladder is constructed in containing satisfying the following: a) a (weak) quasiconvexity property, b) If lies outside a large ball about in the intrinsic metric on , then lies outside a large ball about ) in The quasiconvexity property of ensures control over the hyperbolic geodesic in joining the end-points of ). In particular, if lies outside a large ball about ) in then so

does the geodesic in joining the end-points of ). This guarantees the existence of the Cannon-Thurston map in Theorem 1.1. Scheme of proof of Theorem 1.3: Theorem 1.3 builds on Theorem 1.1 by describing the structur e of the Cannon- Thurston map obtained in [Mj14]. The crux of the proof of Theo rem 1.3 involves
Page 3
ENDING LAMINATIONS AND CANNON-THURSTON MAPS 3 an analysis of the structure of certain specific ladders . The existence and weak quasiconvexity of these ladders was shown in [Mj14], but the analysis (see Steps 2, 3 below) was missing. In fact, even for punctured

torus group s, where the existence of Cannon-Thurston maps was shown by McMullen [McM01], Theo rem 1.3 is new. We now proceed with a step-by-step outline of the proof of The orem 1.3. Step 1) The easy part (Section 3.1) of Theorem 1.3 consists in showing that the end-points of leaves of ending laminations are identified by the Cannon-Thurston map . The essential point is that a leaf of the ending lamination i can be approximated by the lifts to of a sequence of closed curves in whose geodesic realizations exit the relevant end of . We shall refer to this step as the forward direction of

Theorem 1.3. Step 2) The hard part of the proof (Section 4 and Appendix A) co nsists in showing that if the Cannon-Thurston map identifies a pair of points, then they aretheidealend-pointsofaleafoftheendinglaminationor anidealcomplementary polygon. We shall refer to this step as the reverse direction of Theorem 1.3. Bi- infinite geodesics whose end-points are identified by are referred to as CT leaves (cf. Section 3). The proof proceeds by analyzing the structure of the ladder for a CT leaf. The heart of the proof lies in Proposition 4.7 (Asymptotic Qu asigeodesic Rays) which

essentially says that ”vertical” quasigeodesic rays lying on such a ladder are all asymptotic to some point . Further the end-points of are identified with under Step 3) Given Proposition 4.7 there are two ways to complete t he proof of The- orem 1.3: a) Look at the action of ) on the -tree dual to the ending lamination. If there is a CT leaf that is not a leaf of the ending lamination, then we construct a CT leaf (Section 4) whose ideal end-points consist of the attra cting and repelling fixed points ,g for some ). This is a contradiction as is a hyperbolic (loxodromic) element.

This is the approach taken in Section 4. b) Alternately use Proposition 4.7 and a Lemma of Bowditch (L emma 9.2 of [Bow07]) to show that the collection of CT-leaves forms a lam ination. Since the easy direction shows that the ending lamination is containe d in the collection of CT leaves, thisforcesthecollectionofCT-leavestoexactlye qualtheendinglamination. This is the approach taken in Appendix A. Applications: 1)WeprovethefollowingstrengtheningofarigidityTheore mduetoBrock-Canary- Minsky [BCM12]: Theorem 4.10: Let be a closed surface group. Let ) = and ) = be two simply or doubly degenerate

representations of into PSl with limits sets . Suppose that the actions on are topologically conjugate. Then and are quasiconformally conjugate. 2) Theorem 1.3 and its generalization Theorem A.4 are used to prove discrete- ness of commensurators of finitely generated infinite covolu me Kleinian groups in [LLR11] and [Mj11]. 3) Theorems 1.3 and A.4 are extended to arbitrary finitely gen erated Kleinian groups in [Mj10].
Page 4
4 MAHAN MJ WITH SHUBHABRATA DAS Organization of the paper: Section 2 of the paper deals with preliminary concepts and ma terial from [Mj14].

Section 3.1 proves the easy direction of Theorems 1.3 and A.4 : End-points of leaves of the ending lamination are identified by the Cannon-Thurst on map. The argu- ments in Sections 2 and 3 give a unified treatment for surfaces with or without cusps. Section 4 proves the harder direction of Theorem 1.3 f or surfaces without cusps. A slight modification of a fact proven for closed surfa ces (Remark 4.6) will be used for cusped surfaces. We indicate this in Section 4 its elf. Appendix A deals with surfaces with cusps. Acknowledgments: The author would like to thank the

referee(s) for point- ing out errors and omissions and for suggesting corrections . The research for the case without parabolics is supported in part by a DST rese arch grant DyNo. 100/IFD/8347/2008-2009. The research for the case with par abolics is supported in part by a CEFIPRA project grant 4301-1. 2. Preliminaries 2.1. Hyperbolic Metric Spaces. Let ( X,d ) be a hyperbolic metric space and be a subspace that is hyperbolic with the inherited path metr ic . By adjoining theGromovboundaries ∂X and ∂Y to and , oneobtainstheircompactifications and respectively. Let denote

inclusion. Definition 2.1. Let and be hyperbolic metric spaces and be an embedding. A Cannon-Thurston map from to is a continuous extension of The following lemma (Lemma 2.1 of [Mit98]) says that a Cannon -Thurston map exists if and only if for all M > 0 and , there exists N > 0 such that if a geodesic in lies outside an ball around in , then any geodesic in joining the end-points of lies outside the ball around ) in . An equivalent statement is that the Cannon-Thurston map exists if and only if sets of small visual diameter go to sets of small visual diameter. Lemma 2.2. (Lemma 2.1 of

[Mit98] ) Let be an inclusion of hyperbolic metric spaces. A Cannon-Thurston map from to exists if and only if the following condition is satisfied: Given , there exists a non-negative function , such that as , and such that for all geodesic segments lying outside an -ball around , any geodesic segment in joining the end-points of lies outside the -ball around Relative Hyperbolicity and Electric Geometry We refer the reader to [Far98] for terminology and details on relative hyperbolicity and e lectric geometry. Let be a -hyperbolic metric space, and a family of -quasiconvex, separated,

collection of subsets. Recall [Mj14] that electrocution of the collection in means constructing an auxiliary space el ∈H ) with { identified to and { equipped with the zero metric. This is a geometric ‘coning’ construction. Then by work of Farb [Far98], el obtained by electrocuting the subsets in is a ∆ = ∆( δ,C,D ) -hyperbolic metric space.
Page 5
ENDING LAMINATIONS AND CANNON-THURSTON MAPS 5 Now, let = [ a,b ] be a hyperbolic geodesic in and be an electric quasigeodesic without backtracking joining a,b . Starting from the left of , re-

place each maximal subsegment, (with end-points p,q , say) lying within some  { ∈ H ) by a hyperbolic geodesic [ p,q ]. The resulting connected path is called an electro-ambient representative in . Electro-ambient representatives are useful in light of the following. Lemma 2.3. [Mj14] Given δ,C ,D > , there exists such that the following holds: Let X,d be a -hyperbolic metric space, and a family of -quasiconvex, separated, collection of subsets. Let el ,d el denote the electric space obtained from by electrocuting the family . Let = [ a,b be a geodesic in and be an

electro-ambient representative of an electric geodes ic joining a,b . Then α, lies within a bounded distance of each other in el ,d el . Further, lies within a bounded distance of in X,d Partial Electrocution Let be the convex core of a simply (resp. doubly) degenerate 3-ma nifold with cusps. After removing an open neighborhood of cusps we get a m anifold with boundary of the form , where = [0 ) (resp. ), where is a surface with boundary. Surfaces minus open neighborhoods of cusps shall sometimes be referred to as truncated surfaces . Let denote the equivariant collection of horoballs in

covering the cusps of . Let denote minus the interior of the horoballs in . Let denote the collection of boundary horospheres. Then each ∈ H with the induced metric is isometric to a Euclidean product . We shall need to equip each ∈ H is with a new pseudo-metric called the partially electrocuted metric by giving it the product of the zero metric (in the -direction) with the Euclidean metric (in the -direction). The resulting space is quasi-isometric to what one would get by gluing to each the mapping cylinder of the projection of onto the -factor. Let denote the collection of

copies of obtained in this construction and let ( PEY ,d pel ) denote the resulting partially electrocuted space. (See [MP11] for a more general discussion.) We have the follo wing basic Lemma. Lemma 2.4. [MP11] ( PEY ,d pel is a hyperbolic metric space and the sets ∈ J are uniformly quasiconvex. 2.2. Split Geometry. Weshallbrieflyrecalltheessentialaspectsofsplitgeomet ry from [Min10, Mj14]. We shall also need the construction of ce rtain quasiconvex ladder-like sets . Since we shall deal with surfaces with cusps (or punctures) in the Appendix, we give a unified exposition

for surfaces with o r without punctures. If a finite area hyperbolic surface has cusps, we shall remove an open neighborhood of the cusp and denote the resulting truncated surface by . In this subsection therefore will denote a compact surface, possibly with boundary. Split level Surfaces pants decomposition of a compact surface , possibly with boundary, is a disjoint collection of 3-holed spheres ,P embedded in such that is a disjoint collection of non-peripheral annuli in , no two of which are homotopic. Let betheconvexcoreofahyperbolic3-manifoldminusanopenne ighborhood ofthecusp(s).

Thenanyend of issimplydegenerate[Ago04,CG06,Can93]and homeomorphic to [0 ), where is a compact surface, possibly with boundary. A closed geodesic in an end homeomorphic to [0 ) is unknotted if it is
Page 6
6 MAHAN MJ WITH SHUBHABRATA DAS isotopic in to a simple closed curve in { via the homeomorphism. A tube in an end is a regular neighborhood γ,R ) of an unknotted geodesic in Let denote a collection of disjoint, uniformly separated tubes in ends of such that a) all Margulis tubes in belong to for all ends of b) there exists 0 such that the injectivity radius injrad >

for all ∈T Int ) and all ends of Let be a bi-Lipschitz homeomorphism and let (0) be the image of ∈T Int ) in under the bi-Lipschitz homeomorphism . Let ∂M (0) (resp. ∂M ) denote the boundary of (0) (resp. ). Following [Mj14], will be called the model manifold. The metrics on and will be denoted by In [Min10, BCM12], the model manifold refers to with considerable additional structure. In particular it involves the decomposition of (0) into pieces of the form and where and refer to a sphere with 4 holes and a torus with one hole respectively. To distinguish between t he

model manifold in [Min10, BCM12] and that in this paper we shall refer to the for mer as the Minsky model. It should be pointed out that Minsky model was constru cted by Minsky in [Min10] and proven to be bi-Lipschitz homeomorphic to the hy perbolic manifold by Brock-Canary-Minsky in [BCM12]. Let ( Q,∂Q ) be the unique hyperbolic pair of pants such that each compon ent of ∂Q has length one. will be called the standard pair of pants. An isometrically embedded copy of ( Q,∂Q ) in ( (0) ,∂M (0)) will be said to be flat Definition 2.5. split level surface

associatedtoapantsdecomposition ,Q of in (0) is an embedding ,∂Q (0) ,∂M (0)) such that 1) Each ,∂Q ) is flat 2) extends to an embedding (also denoted ) of into such that the interior of each annulus component of ) lies entirely in ∈T Int )). Let denote the union of the collection of flat pairs of pants in the image of the embedding The class of all topological embeddings from to that agree with a split level surface associated to a pants decomposition ,Q on will be denoted by [ ]. We define a partial order on the collection of split level surfaces in

an end of as follows: if there exist ], = 1 2, such that ) lies in the unbounded component of ). A sequence of split level surfaces is said to exit an end if i < j implies and further for all compact subsets , there exists L > 0 such that for all Definition 2.6. A curve in is -thin if the core curve of the Margulis tube has length less than or equal to . A tube ∈ T is -thin if its core curve is -thin. A tube ∈ T is -thick if it is not -thin. A curve is said to split a pair of split level surfaces and i < j ) if occurs
Page 7
ENDING LAMINATIONS AND CANNON-THURSTON

MAPS 7 as a boundary curve of both and The collection of all -thin tubes is denoted as . The union of all -thick tubes along with (0) is denoted as ). Definition 2.7. A pair of split level surfaces and i < j ) is said to be separated if a) for all x,S b)Similarly, for all x,S Definition 2.8. An -bi-Lipschitz split surface in ) associated to a pants decomposition ,Q of and a collection ,A of complementary annuli (not necessarily all of them) in is an embedding such that 1) the restriction ,∂Q (0) ,∂M (0)) is a split level surface 2) the restriction ) is an -bi-Lipschitz

embedding. 3) extends to an embedding (also denoted ) of into such that the interior of each annulus component of )) lies entirely in ∈T Int )). Note: The difference between a split level surface and a split surfa ce is that the latter may contain bi-Lipschitz annuli in addition to flat pa irs of pants. We denote split surfaces by to distinguish them from split level surfaces Let denote the union of the collection of flat pairs of pants and bi -Lipschitz annuli in the image of the split surface (embedding) Theorem 2.9. [Mj14, Theorem4.8] Let N,M,M (0) ,S,F be as above

and an end of . For any less than the Margulis constant, let ) = ) : injrad . Fix a hyperbolic metric on such that each component of ∂S is totally geodesic of length one. There exist , and a sequence of -bi- Lipschitz, -separated split surfaces exiting the end of such that for all , one of the following occurs: (1) An -thin curve splits the pair ( +1 , i.e. splits the associated split level surfaces ,S +1 , which in turn form an -thin pair. (2) there exists an -bi-Lipschitz embedding : ( [0 1] ∂S [0 1]) M,∂M such that { and +1 { Finally, each -thin curve in

splits at most split level surfaces in the sequence A model manifold all of whose ends are equipped with a collection of exiting split surfaces satisfying the conclusions of Theorem 2.9 is said to be equipped with weak split geometry structure. Pairs of split surfaces satisfying Alternative (1) of Theor em 2.9 will be called an -thin pair of split surfaces (or simply a thin pair if is understood). Similarly, pairs of split surfaces satisfying Alternative (2) of Theor em 2.9 will be called an -thick pair (or simply a thick pair) of split surfaces. Definition 2.10. Let ( +1 be a thick pair

of split surfaces in . The closure of the bounded component of ( +1 between +1 will be called a thick block.
Page 8
8 MAHAN MJ WITH SHUBHABRATA DAS Note that a thick block is uniformly bi-Lipschitz to the prod uct [0 1] and that its boundary components are +1 Definition 2.11. Let ( +1 be an -thin pair of split surfaces in and be the collection of -thin Margulis tubes that split both +1 . The closure of the union of the bounded components of (( +1 )) between +1 will be called a split block. Equivalently, the closure of th e union of the bounded components of ( +1 between +1 is a

split block. Each connected component of a split block is a split component. Remark 2.12. [Mj14, Remark 4.12] For each lift of a split component of a split block of , there are lifts of -thin Margulis tubes that share the boundary of in . Adjoining these lifts to we obtain extended split components . Let denote the collection of extended split components in Denote the collection of split components in by . Let ) denote the lift of ) to . Then the inclusion of ) into gives a quasi-isometry between ) and M, ) equipped with the respective electric metrics. This follows from the last assertion of

Theorem 2.9. The electric metric on M, ) is called the graph-metric [Mj14, Section 4.3] and is denoted by . The electric space will be denoted as ( M,d ). The electric metric on M, ) is quasi-isometric to the electric metric on M, ), again by the last assertion of Theorem 2.9. The electric sp ace will be denoted as ( M,d ). Definition 2.13. Let and is said to be -graph quasiconvex if for any hyperbolic geodesic joining a,b lies inside X,d ⊂ E M, For a split component in a manifold, define CH ) = CH )), where CH ) is the convex hull of in , provided the ends of have no cusps,

i.e. . Else define CH ) to be the image under of CH ) minus cusps. Further, in order to ensure hyperbolicity of the univ ersal cover, we partially electrocute the cusps of (cf. Theorem 2.4). Then∆-graphquasiconvexityof isequivalenttotheconditionthat dia CH )) is bounded by = (∆) as any split component has diameter one in ( M,d ). We recall the following from [Mj14]. Lemma 2.14. [Mj14, Lemma 4.16] Let be a simply degenerate end of a simply or doubly degenerate hyperbolic 3-manifold homotopy equivalent to a surface and equipped with a weak split geometry model . For a split

component contained in , let be a lift to . Then there exists such that the convex hull of minus cusps lies in a -neighborhood of in Proposition 2.15. [Mj14, Proposition 4.23] For a split component, is uni- formly graph-quasiconvex in , i.e. there exists such that dia CH ))) for all incompressible split components We summarize the conclusions of the above propositions belo w. Definition 2.16. A model manifold of weak split geometry is said to be of split geometry if
Page 9
ENDING LAMINATIONS AND CANNON-THURSTON MAPS 9 (1) Each split component is quasiconvex (not necessarily

uniformly) in the hyperbolic metric on (2) Equip with the graph-metric obtained by electrocuting (extended) split components . Then the convex hull CH of any split component has uniformly bounded diameter in the metric Hence by Lemma 2.14 and Proposition 2.15 we have the followin g technical Theorem of [Mj14]. Theorem 2.17. [Min10, BCM12] [Mj14, Theorem 4.32] Any simply or doubly degenerate hyperbolic 3-manifold homotopy equivalent to a surface is bi-Lipschitz homeomorphic to a Minsky model and hence to a model of split ge ometry. 2.2.1. Ladders: For details on the construction of ladders, see

[Mj14, Secti on 5]. Note that after welding the boundary components of together in a split block, we obtain a bounded geometry surface in wel . Thus is the connected bounded geometry surface obtained from by equipping it with the quotient topology dictated by welding. For convenience of notation, we redesi gnate this surface Intheweldedmodelmanifold wel , wethusobtainasequenceofboundedgeometry surfaces exiting the end(s). The region between and +1 is either a thick block or a split block. From a geodesic { } we constructed in [Mj14] a ‘hyperbolic ladder wel ) such that is an

electro-ambient quasigeodesic in the (path) electric metric on induced by the graph metric on +1 is constructed inductively from (in [Mj14] or [Mj05]) by ‘flowing up’ in the block . More precisely, has a natural product structure and is bounded by and +1 . Given joining ,q , there exist points +1 ,q +1 +1 lying vertically above ,q respectively. +1 is the electro-ambient geodesic in +1 (equipped with the electric metric) joining +1 ,q +1 We also constructed a large-scale retract → L such that the restriction of to { is, roughly speaking, a nearest-point retract of {

onto in the (path) electric metric on We have the following basic theorem from [Mj14] Theorem 2.18. [Mj14, Theorem 5.7] There exists C > such that for any geodesic { } , the retraction → L satisfies: Then ( )) Cd x,y )+ 2.2.2. qi Rays: We also have the following from [Mj14]. Lemma 2.19. [Mj14, Lemma 5.9] There exists such that for there exists with ,x . Similarly there exists +1 +1 with ,x +1 . Hence, for all and , there exists a -quasigeodesic ray such that ⊂ L for all and ) = Further, by construction of split blocks, ,S ) = 1. Therefore inductively, ,S ) = . Hence

,x ≥ | . By construction, ,x Hence, given the sequence of points ,n ∪{ (for simply degenerate groups) or (for totally degenerate groups) with gives by Lemma 2.19
Page 10
10 MAHAN MJ WITH SHUBHABRATA DAS above, a quasigeodesic in the -metric. Such quasigeodesics shall be referred to as -quasigeodesic rays 3. Laminations 3.1. Ideal points are identified by Cannon-Thurston Maps. We would like to know exactly which points are identified by the Cannon-Thu rston map, whose existence is ensured by Theorem 1.1. Let denote inclusion. Let be the continuous extension of

to the disk = ( ) in Theorem 1.1. Let ∂i denote the restriction of to the boundary As mentioned in the introductory Section 1.2, we shall first p rove the forward direction of Theorem 1.3. Proposition 3.1 below shows that the existen ce of a Cannon-Thurston map automatically guarantees that end-po ints of leaves of the ending lamination are identified by the Cannon-Thurston map Proposition 3.1. Let u,v be either ideal end-points of a leaf of an ending lami- nation, or ideal boundary points of a complementary ideal po lygon. Then ∂i ) = ∂i Proof. ( cf. Lemma 3.5 of

[Mit97]. See also [Mj14].) We shall use two f acts in the proof: 1) the fact (due to Bonahon [Bon86] and Thurston [Thu80]) tha t surface groups are tame and that for there exist simple closed curves on whose geodesic realizations exit the end. 2) the fact (due to Thurston [Thu80] Ch. 9) that the sequence o f simple closed curves converges to the ending lamination in the space of measured l aminations. It follows that after lifting to the universal cover, any lea f of the ending lamination is a Chabauty topology limit of bi-infinite geodesics (lifts of ). Since any end of is geometrically

tame [Thu80], there exists such that there exists a sequence of closed geodesics with length at most exiting the end. We shall refer to such geodesics as ‘bounded geodesics . Let be geodesics in the intrinsic metric on the base surface ( = ) freely homotopic to We can assume further [Bon86] that ’s are simple. Join to by the shortest geodesic in connecting the two curves. For any leaf of the ending lamination, we have a subsequence of the ’s whose Hausdorff limit in contains . Abusing notation slightly let us denote the sub- sequence as . In the universal cover, we obtain segments fi

which are finite segments whose end-points are identified by the coveri ng map We also assume that is injective restricted to the interior of fi ’s mapping to Similarly there exist segments fi which are finite segments whose end-points are identified by the covering map . We also assume that is injective restricted to the interior of fi ’s. The finite segments fi and fi are chosen in such a way that there exist lifts , joining end-points of fi to corresponding end-points of fi . The union of these four pieces looks like a trapezium (See fig ure below, where

we omit subscripts for convenience).
Page 11
ENDING LAMINATIONS AND CANNON-THURSTON MAPS 11 Figure 2: Trapezium Next, given any lift of the leaf to , we may choose translates of the finite segments fi (under the action of )) appropriately, such that fi converge to in (the Hausdorff/Chabauty topology on closed subsets of) . For each fi , let fi fi where denotes with orientation reversed. Then fi ’s are uniform hyperbolic quasigeodesics in (since fi is short). If the translates of fi we are considering have end-points lying outside large balls around a fixed refe

rence point , it is easy to check that fi ’s lie outside large balls about in At this stage we invoke the existence theorem for Cannon-Thu rston maps, The- orem 1.1. Since fi ’s converge to and there exist uniform hyperbolic quasi- geodesics fi , joining the end-points of fi and exiting all compact sets, it follows that ∂i ) = ∂i ), where a,b denote the boundary points of Hence if we define u,v to be equivalent if they are the end-points of a leaf of the ending lamination, then the transitive closure of this rela tion has as elements of an equivalence class a) either ideal

end-points of a leaf of a lamination, b) or ideal boundary points of a complementary ideal polygon c) or a single point in which is not an end-point of a leaf of a lamination. Definition 3.2. Let be a finitely presented group acting on a hyperbolic space with quotient . Let be a 2-complex with fundamental group , and be a map inducing an isomorphism of fundamental groups. Then lifts to . A bi-infinite geodesic in will be called a leaf of the abstract ending lamination for , if 1) there exists a set of geodesics in exiting every compact set 2) there exists a set of geodesics

in with freely homotopic to 3) there exist finite lifts of in such that the natural covering map Π : is injective away from end-points of 4) converges to in the Chabauty topology Proposition 3.1 and its proof readily generalize to Proposition 3.3. Suppose is hyperbolic and extends to a Cannon- Thurston map on boundaries. Let u,v be end-points of a leaf of an abstract ending lamination. Then ∂i ) = ∂i To distinguish between the ending lamination and bi-infinit e geodesics whose end-points are identified by ∂i , we make the following definition.


Page 12
12 MAHAN MJ WITH SHUBHABRATA DAS Definition 3.4. CT leaf CT is a bi-infinite geodesic whose end-points are identified by ∂i An EL leaf EL is a bi-infinite geodesic whose end-points are ideal boundar y points of either a leaf of the ending lamination, or a complementary ideal polygon. Then to prove the main theorem 1.3 it remains to show that CT leaf is an EL leaf 3.2. Leaves of Laminations. Our first observation is that any semi-infinite geo- desic (in the hyperbolic metric on ) contained in a CT leaf in the base surface { }

{ has infinite diameter in the graph metric restricted to { , i.e. the induced path metric on { . This follows from the following somewhat stronger assertion. Lemma 3.5. Given , there exists such that if and is a bi-infinite geodesic in the intrinsic metric on , whose end-points are identified by the Cannon-Thurston map, then for any split component dia hyp Proof. Suppose not. Then there exist split components , such that dia hyp )) where dia hyp denotes diameter in the hyperbolic metric on . Acting on by elements of the surface group ), we may assume

that there exists a sequence of segments such that is approximately centered about a fixed origin 0 in a fixed lift of a fixed split component , i.e. pass uniformly close to 0 and end-points of are at distance from 0. This is possible since contains finitely many split blocks. Since is quasiconvex, it follows that the ’s are uniform quasigeodesics in Hence, the sequence converges to a bi-infinite quasigeodesic in the Chabauty topology. Since the set of CT leaves are closed in the Chabauty topology, it follows that is a CT leaf But, this is a contradiction, as we

have noted already that is a quasigeodesic. Corollary 3.6. CT leaves have infinite diameter Let  { be a semi-infinite geodesic (in the hyperbolic metric on ) contained in a CT leaf . Then dia is infinite, where dia denotes diameter in the graph metric restricted to Proof. Put = 1 in Lemma 3.5. Using Lemma 3.5, we shall now show: Proposition 3.7. There exists a function as such that the following holds: Let be a CT leaf. Also for p,q , let pq denote a geodesic in M,d joining p,q . If ,b be such that 0) 0) , then 0) where denotes the hyperbolic metric on and the graph

metric. Proof. Suppose not. Let and denote the ideal end points of . Then there exists 0, such that 0) . That is, there exist such that (0 ,p . Due to the existence of a Cannon-Thurston map
Page 13
ENDING LAMINATIONS AND CANNON-THURSTON MAPS 13 in the hyperbolic metric (Theorem 1.1), we may assume that (0 ,p (in the hyperbolic metric). Then the hyperbolic geodesic ,p passes through at most split blocks (cf. Definition 2.11) for every . Let and . Then ,p . But since , the Cannon-Thurston map identifies , ,p See Figure below. Figure 3: Cannon-Thurston in the Graph Metric

Also, ,p ,p , ,p ,p , and at least one of the above two ( ,p , , say) must pass close to 0. Then , is a CT leaf . But ,p lies in a -neighborhood of 0 in the graph-metric , contradicting Lemma 3.5 above. 4. Closed Surfaces In this section will denote a closed surface. As mentioned in the introducto ry Section 1.2, we shall now proceed to prove the reverse direction of Theorem 1.3. The aim of this Section is to show that a CT leaf is an EL leaf. 4.1. Geodesic Laminations and -trees. For a discussion of geodesic lamina- tions (or simply laminations as we shall call them), we refer the reader to

[PH92], [CEG87], [Thu80], [CB87]. For a discussion on dual -trees, see [Sha91]. The space of filling laminations which we denote FL are the measure classes of measured laminations Λ for which all complementary regio ns of the support are simply connected. The quotient of FL by forgetting the measures will be denoted EL and is the space of ending laminations . It is a well-known fact [Thu80, Min10] that ending laminations have no simple close d leaves. A useful fact is that such laminations are minimal, i.e. the closure (in th e Hausdorff topology) of any of its leaves is the

whole lamination. We can identify a minim al lamination Λ with a closed invariant (under )) subset of the set of unordered pairs in ∆) , where ∆ denotes the diagonal and is the relation identifying a,b ) with ( b,a ). Lemma 4.1. Let be a minimal geodesic lamination on a surface . Let be an embedded (closed) interval in transverse to . Let denote the union of all lifts of leaves of to the universal cover . Let denote the union of all lifts of to Define two leaves of to be equivalent if both of them intersect the same component
Page 14
14 MAHAN MJ WITH

SHUBHABRATA DAS of . Then the limit set of any connected component of the transit ive closure of this relation contains a pair of poles and for some element Proof. Let be the -tree dual to Λ. Let be a fixed lift of to . Then projects to an embedded non-trivial interval (also called ) in under the quotient map that identifies leaves of Λ to points. The orbit of under ) acting on is a forest, in fact a sub-forest of Let be the connected component of containing . If \ T , then ⊂ T . Hence \T , and we finally have that is invariant under for all integers . This

shows that contains the pole corresponding to the infinite order element Thus we need finally the existence of a as in the previous paragraph. It suffices to show that for any non-trivial , there exists ) such that gI But this follows from minimality of Λ, using the fact that eac h leaf is dense in Λ, and hence that there exists ) such that gI and are transverse to a common leaf of Λ. 4.2. Rays Contained in Ladders. Definition 4.2. Let X,Y,Z be geodesically complete metric spaces such that is said to coarsely separate into and if (1) (2) (3) For all ,

there exist and such that ,Y and ,Y (4) There exists such that for all and any geodesic in joining ,y passes through a -neighborhood of Let be any bi-infinite geodesic in . Let be the ladder corresponding to as in Theorem 2.18. We now fix a quasigeodesic ray as in Lemma 2.19, and consider a translate passing through ⊂ L , i.e. ) = . Let ⊂ L Each ) cuts into two pieces and with ideal boundary points i, , i, respectively. We shall show that coarsely separates into and , where and (resp. ) is the segment of joining ) to the ideal end-point i, (resp. i, We need to

repeatedly apply Theorem 2.18 to prove the above as sertion. Given , we construct two hyperbolic ladders and −0 , obtained by joining the points ) to the ideal end-points i, , i, respectively, of  { Then and −0 are -quasiconvex (in the graph metric ) by Theorem 2.18. Further, ) = ) by definition of . Hence, −0 ) = ) = Further,
Page 15
ENDING LAMINATIONS AND CANNON-THURSTON MAPS 15 ∪L \L and there exists a (independent of ,h, ) such that ,r are all quasiconvex. Criterion (3) of Definition 4.2 in this context is given by Lem ma 3.6 : CT

leaves have infinite diameter To prove that separates into , we need to show first: Lemma 4.3. For all , there exists such that if ∈ L ,q ∈ L with p,q , then there exists such that p,z and q,z Proof. Let denote the sheetwise retract of Theorem 2.18 onto . Then ) = and ) = for all ∈ L Hence ) = ) = for some and some Therefore, by Theorem 2.18 again, q,z Cd p,q ) = CK Choosing CK (and using the triangle inequality for p,q,z ) the Lemma follows. We are now in a position to prove: Theorem 4.4. coarsely separates into Proof. We have already shown ∪L \L and there

exists a (independent of ,h, ) such that ,r are all quasiconvex. Criterion (3) of Definition 4.2 is given by Lemma 3.6. Finally given ∈ L and ∈ L , let uv be the geodesic in ( M,d ) joining u,v . Then uv ) is a ”dotted quasigeodesic”, i.e. there is a sequence of poi nts ,p , where ,p +1 (and the constant is obtained from Theorem 2.18). Further, ∈ L ,p ∈ L and ∈ L for all . Therefore there exists such that ∈ L ,p +1 ∈ L , with ,p +1 . Hence, by Lemma 4.3, there exists 0 such that there exists with ,z and +1 ,z Finally, byTheorem2.17, ( M,d

)ishyperbolic, andthereforethe”dottedquasi- geodesic ,p lies in a uniformly bounded neighborhood of the geodesic uv . That is, there exists 0 such that for all ∈ L and ∈ L
Page 16
16 MAHAN MJ WITH SHUBHABRATA DAS the geodesic uv in ( M,d ) joining u,v passes through a -neighborhood of This proves (4) in Definition 4.2 and hence we conclude that coarsely into We shall have need for the following Proposition, whose proo f is exactly along the lines of Theorem 4.4 above. Proposition 4.5. Let be two bi-infinite geodesics on such that Then \L contains a

quasigeodesic ray coarsely separating both and Remark 4.6. Proposition 4.5 generalizes readily to cusped surfaces to show that if are two bi-infinite geodesics on such that , then \L contains a quasigeodesic ray To see this it suffices to note that if , then for all . Hence we may construct a quasigeodesic ray contained in both and OnelastPropositiontobeusedintheproofofTheorem1.3ist hefollowingwhich says in particular that any two quasigeodesic rays lying on are asymptotic with respect to the graph metric .. Proposition 4.7. Asymptotic Quasigeodesic Rays Given there exists such that

if is a CT-leaf then there exists satisfying the following: If and are -quasi-geodesic rays contained in then there exists such that 1) ∂i ) = ∂i as , for = 1 2) ,r )) for all Proof. By Proposition 3.7, we find that if ,b such that ,b converge to ideal points , (denoted , for convenience), then ) leaves large balls about 0. More precisely there exists as such that ) lies outside the ball about 0. Also, by Theorem 4.4 above, each coarsely separates . Hence passes close to ) for some , where as . We conclude that any such converges on to the same point as ) = ). This proves

(1). In particular any two quasigeodesic rays lying on are asymptotic with respect to the graph metric . This proves (2). 4.3. Main Theorem for Simply Degenerate Groups. We are now in a posi- tion to prove the main Theorem 1.3 of this paper for closed sur faces. For ease of exposition we shall deal with the simply degenerate case firs t and then indicate the additional niceties for doubly degenerate groups. Recall t hat for a simply degen- erate manifold , where = [0 ). For a totally degenerate manifold = ( ) and it is the presence of two ends, positive and negative, th at ne- cessitates

further care. The split level surfaces are index ed by 0 for a simply degenerate manifold and by for a totally (doubly) degenerate manifold. For a doubly degenerate group, there will be two ending laminations, one for each end and a slight modification of the proof below will be necess ary to identify and distinguish these. The constructions of ladders and blocks are otherwise identical in both cases.
Page 17
ENDING LAMINATIONS AND CANNON-THURSTON MAPS 17 Theorem 4.8. Let ∂i ) = ∂i for a,b be two distinct points that are identified by the Cannon-Thurston map

corresponding to a sim ply degenerate closed surface group (without accidental parabolics). Then a,b are either ideal end-points of a leaf of the ending lamination (in the sense of Thurston), or ideal boundary points of a complementary ideal polygon. Further, if a,b are either ideal end-points of a leaf of a lamination, or ideal boundary points of a comple mentary ideal polygon, then ∂i ) = ∂i Proof. The second statement has been shown in Proposition 3.1. To prove the first statement, let ∂i ) = ∂i ) for a,b . Then ( a,b ) = is a CT-leaf. Suppose and are

intersecting CT leaves, i.e. ∂i ) = ∂i ) and ∂i ) = ∂i ). As before, let and be intersections of the ladders and with the horizontal sheets. Then ) = is a quasigeodesic ray by Proposition 4.5. By Proposition 4.7, ) converges to a point on as such that ∂i ) = ∂i ) = ∂i ) = ∂i ). Hence the Cannon-Thurston map identifies the endpoints of any two intersecting CT leaves and If possible, suppose that the CT leaf is not an EL-leaf. Then intersects the ending lamination transversely (since the ending laminati on is a filling lamination

without any closed leaves) and there exist EL-leaves for which . By Proposition 3.1, each such is a CT-leaf. Hence, by the previous paragraph, the Cannon-Thurston map ∂i identifies the end points of with the endpoints of each such EL-leaf for which . Let ) denote this common image under ∂i Since is not an EL-leaf, it contains a non-trivial geodesic subseg ment trans- verse to the ending lamination. Then the common image (under ∂i ) of end-points of all EL leaves intersecting transversely is By Lemma 4.1 ( ∂i ) contains a pair of poles ,g for some ). This is

because the equivalence class defined by as in Lemma 4.1 consists of pairs of points all of which are identified (under ∂i ) with This is a contradiction as a pair of poles forms the end-point s of a quasigeodesic in . We conclude that must be an EL-leaf. 4.4. Modifications for Totally Degenerate Groups. Weelaborateonthemod- ifications indicated in the first paragraph of Section 4.3, to pass from the simply degenerate case to the totally degenerate case. The constru ction of the ‘hyperbolic ladder as in the discussion preceding Theorem 2.18 is done with inde

xing set in place of . In particular the quasigeodesic ray of Lemma 2.19 is replac ed by a bi-infinite quasigeodesic . However, as a hyperbolic metric space has two boundary points + and . Correspondingly we have two ending laminations and . The easy direction of Theorem 1.3 given by Proposition 3.1 t hen goes through verbatim to show that CT We need to find a way of distinguishing the + and directions in CT . To implement this, note that the discussion preceding Proposi tion 4.7 shows that if CT , i.e. ∂i ) = ∂i ), then we have a bi-infinite quasigeodesic such

that ∂i ) = ∂i ) = ), where is either + or . Define CT CT (resp. CT CT ) to be the collection of CT -leaves whose endpoints
Page 18
18 MAHAN MJ WITH SHUBHABRATA DAS are identified in the + (resp. ) direction, i.e. = + (resp. ). Then the forward direction of Theorem 1.3 given by Proposition 3.1 sh ows that CT and CT . Since both ending laminations and are individually filling arational minimal laminations, the proof of Theorem 4.8 (th e reverse direction for simply degenerate groups) now shows that in fact = CT and = CT 4.5. Application: Rigidity. In

[BCM12], Brock-Canary-Minsky prove the fol- lowing Rigidity Theorem. Theorem 4.9. Let be a closed surface group. If and are two discrete faithful representations of into PSl that are conjugate by an orientation-preserving homeomorphism of , then and are quasiconformally conjugate. We strengthen this by weakening the hypothesis of Theorem 4. 9 to a topological conjugacy only on limit sets (rather than all of ). Theorem 4.10. Let be a closed surface group. Let ) = and ) = be two simply or doubly degenerate representations of into PSl with limits sets . Suppose that the actions on are

topologically conjugate. Then and are quasiconformally conjugate. Proof. We first deal with the simply degenerate case. By Theorem 1.3, the pre- images of the Cannon-Thurston maps ∂i and ∂i from ∂G (= ) to Λ or are given by end-points of leaves of the ending lamination (or id eal points of comple- mentary polygons) whenever ∂i and ∂i are non-injective. Thus the action on pulls back to a equivariant homeomorphism ∂G ∂G taking the ending lamination of to that of . Re-marking by an isomorphism of if necessary, we can ensure that the

homeomorphism be the ident ity on ∂G . Hence the ending laminations of and are the same. In the doubly degenerate case, the same argument shows that t he pairs of ending laminations for and are the same. Hence if and are doubly degenerate, they have the same end-invariants. By the Ending Lamination Theorem [BCM12], and are conformally conjugate In the simply degenerate case, the conformal structures cor responding to the geometrically finite ends for and are quasiconformal deformations of each other (since the quotient of the domain of discontinuity is a connected finite

volume Riemann surface). Since the ending laminations of and are the same, it follows therefore that the quotient manifolds are bi-Lipschitz hom eomorphic by the Ending LaminationTheorem[BCM12]. Hence and are quasiconformally conjugate Appendix A (by Shubhabrata Das and Mahan Mj) Surfaces with Cusps We now deal with surfaces with cusps. will denote a finite volume hyperbolic surface with cusps. will denote a truncated surface, i.e. minus an open neighborhood of the cusps. The arguments in this Section can be easily adapted to The work in this Appendix forms part of SD’s PhD thesis

written und er the supervision of MM. The proof given here was discovered jointly considerably after th e work on the earlier Sections was completed. Hence we have retained both approaches.
Page 19
ENDING LAMINATIONS AND CANNON-THURSTON MAPS 19 furnish a slightly different proof of Theorem 1.3 for surface s without cusps. As in the previous Section we deal with the case of simply degenera te groups first. Equivalence Relations on Suppose that a group acts on preserving a closed equivalence relation . An example of such a relation comes from a lamination , where two points

on are declared equivalent if they are end-points of a leaf of . The equivalence relation is obtained as the transitive closure of this relation. Definition A.1. [Bow07] Two disjoint subsets, P,Q are linked if there exist linked pairs, x,y } and z,w } is unlinked if distinct equivalence classes are unlinked. The following Lemma due to Bowditch give us a way of recognizi ng relations coming from laminations. Lemma A.2. (Lemma 9.2 of [Bow07] ) Let be a non-empty closed unlinked -invariant equivalence relation on . Suppose that no pair of fixed points of any loxodromic are

identified under . Then there is a unique complete perfect lamination, , on such that Let CT denote the equivalence relation on induced by the Cannon-Thurston map for a simply degenerate punctured surface group (cf. The orem 1.1). Let denote the ending lamination. By Proposition 3.1 pairs of en d-points of leaves of are contained in CT . Hence, for simply degenerate groups, it suffices to show that CT is induced by a lamination since no other lamination can prop erly contain Λ. By Lemma A.2 it suffices to show that CT is unlinked. The next Proposition is the analogue of

Proposition 4.7 for cusped surfaces. Proposition A.3. Let be an inclusion of the universal cover of a punctured surface into the universal cover of the convex co re of a simply degenerate 3-manifold. Let ∂i be the associated Cannon-Thurston map. If is a CT-leaf in the corresponding ladder, and ⊂ L a qi ray, then there exists such that ∂i ) = ∂i as Proof. We first observe that both end-points , of the CT leaf cannot be parabolics. For then they would have to be base points of different horoballs in as they correspond to different lifts of the cusp(s) of

Case 1: Both , are non-parabolic. The proof of Proposition 4.7 goes through in this context mut atis mutandis. Case 2: Exactly one of , is a parabolic. Without loss of generality assume that is a parabolic. Let be the horoball in based at ∂i ) and let be the horosphere boundary of . Let be the point of intersection of with . For p,q , ( p,q and pq will denote respectively geodesics in ( ,d ) and ( ,d ). Choose a sequence of points ,b such that and Then by the existence of Cannon-Thurston maps for (Theorem 1.1) it follows that there exists a function as such that ( ,b lies outside .

Hence, if = ( ,b then ,o ) and the geodesic subsegment ( ,b lies outside
Page 20
20 MAHAN MJ WITH SHUBHABRATA DAS Let be the complement of open horoballs and be the graph metric on obtained after first partially electrocuting horospheres ( cf. Section 2.1). By Lemma 2.3 ( ,b and lie in a bounded neighborhood of each other in ( N,d ). The -distance o,q )isequaltothenumberofverticalblocksbetween and . But implies in . Hence o,q as By Corollary 3.6, o,b as . Hence by Proposition 3.7 there exists a function as such that lies outside N,d ). Now recall that → L is a coarse

Lipschitz retract by Theorem 2.18. Hence ⊂ L is a uniform quasigeodesic in ( N,d ). Further, since belongs to and since essentially fixes the horosphere , it follows that ( ,q 1. Also ) = . Therefore there exists a function as such that ] lies outside N,d ). Next, since \L and lie on different sides of the qi ray ⊂ L it follows that there exists such that ,r ) is uniformly bounded. Also there exists ,b such that ,t ) and hence ,r ) is uni- formly bounded. Since ,b it follows that ∂i ) = ∂i ). Since ,r ) is uniformly bounded, there exists such that ,s ) is

uniformly bounded and therefore ,s are separated by a uniformly bounded number of split com- ponents. By uniform graph quasiconvexity of split componen ts (Theorem 2.17) it follows that ∂i ) = ∂i ). Finally if denotes the part of the ray ‘above , (i.e. [ )) then joining points of in successive blocks by hyperbolic geodesics we obtain an el ectro- ambient quasigeodesic . By Lemma 2.3 there exist hyperbolic geodesics m,n joining ,r ) for m > n and contained in a bounded neighborhood of in . Hence ∂i ) = ∂i ) as We are now in a position to prove the analogue of Theorem 1.3

fo r surfaces with cusps. Theorem A.4. Let ∂i ) = ∂i for a,b be two distinct points that are identified by the Cannon-Thurston map corresponding to a sim ply degenerate sur- face group (without accidental parabolics). Then a,b are either ideal end-points of a leaf of the ending lamination, or ideal boundary points of a complementary ideal polygon. Further, if a,b are either ideal end-points of a leaf of a lamination, or ideal boundary points of a complementary ideal polygon, the ∂i ) = ∂i Proof. The second statement has been shown in Proposition 3.1. To prove the

first statement, it suffices to show that CT is unlinked. Suppose now that and are intersecting CT leaves, i.e. ∂i ) = ∂i ) and ∂i ) = ∂i ). Consider the ladders and . Let be a quasigeodesic ray as per Remark 4.6. By Proposition A.3, converges to a point on such that ∂i ) = ∂i ) = ∂i ) = ∂i ). Hence if a,b c,d } ∈ R CT , then either a,b,c,d are all mutually related in CT , or a,b c,d are unlinked. By Lemma A.2, CT is induced by a lamination CT . By Proposition 3.1, the ending
Page 21
ENDING LAMINATIONS AND

CANNON-THURSTON MAPS 21 lamination EL is contained in CT . Since EL is filling and arational, it follows that EL = CT The modifications necessary to pass from the simply degenera te case to the totally degenerate case are exactly as in the case of surface s without cusps. References [Ago04] I. Agol. Tameness of hyperbolic 3-manifolds. preprint, arXiv:math.GT/0405568 , 2004. [BCM12] J. F. Brock, R. D. Canary, and Y. N. Minsky. The Classificatio n of Kleinian surface groups II: The Ending Lamination Conjecture. Ann. of Math. 176 (1), arXiv:math/0412006 , pages 1–149, 2012.

[Bon86] F. Bonahon. Bouts de varietes hyperboliques de dimension 3. Ann. of Math. (2) 124 pages 71–158, 1986. [Bow07] B. H. Bowditch. The Cannon-Thurston map for punctured surfa ce groups. Math. Z. 255 , pages 35–76, 2007. [Can93] R. D. Canary. Ends of hyperbolic 3 manifolds. J. Amer. Math. Soc. , pages 1–35, 1993. [CB87] A. Casson and S. Bleiler. Automorphisms of Surfaces after Nielsen an d Thurston. Lon- don Math. Soc. Student Texts, Cambridge , 1987. [CEG87] R. D. Canary, D. B. A. Epstein, and P. Green. Notes on Notes of T hurston. in Analytical and Geometric Aspects of Hyperbolic Spaces ,

pages 3–92, 1987. [CG06] D. Calegari and D. Gabai. Shrink-wrapping and the Tami ng of Hyperbolic 3-manifolds. J. Amer. Math. Soc. 19, no. 2 , pages 385–446, 2006. [CT85] J. Cannon and W. P. Thurston. Group Invariant Peano Curv es. preprint, Princeton 1985. [CT07] J. Cannon and W. P. Thurston. Group Invariant Peano Curv es. Geom. Topol. 11 , pages 1315–1355, 2007. [Far98] B. Farb. Relatively hyperbolic groups. Geom. Funct. Anal. 8 , pages 810–840, 1998. [LLR11] C. J. Leininger, D. D. Long, and A. W. Reid. Commensurators of non-free finitely generated Kleinian groups. Algebr. Geom. Topol.

11 arXiv:math 0908.2272 , pages 605 624, 2011. [McM01] C. T. McMullen. Local connectivity, Kleinian groups and geodesi cs on the blow-up of the torus. Invent. math. , 97:95–127, 2001. [Min94] Y. N. Minsky. On Rigidity, Limit Sets, and End Invariant s of Hyperbolic 3-Manifolds. J. Amer. Math. Soc. 7 , pages 539–588, 1994. [Min10] Y. N. Minsky. The Classification of Kleinian surface groups I : Models and bounds. Ann. of Math. 171 (1) , pages 1–107, 2010. [Mit97] M. Mitra. Ending Laminations for Hyperbolic Group Ex tensions. Geom. Funct. Anal. , pages 379–402, 1997. [Mit98] M. Mitra.

Cannon-Thurston Maps for Hyperbolic Group E xtensions. Topology 37 , pages 527–538, 1998. [Mj05] M. Mj. Cannon-Thurston Maps for Surface Groups: An Exposi tion of Amalgamation Geometry and Split Geometry. preprint, arXiv:math.GT/0512539 , 2005. [Mj10] M. Mj. Cannon-Thurston Maps for Kleinian Groups. preprint, arXiv:math 1002.0996 2010. [Mj11] M. Mj. On Discreteness of Commensurators. Geom. Topol. 15, arXiv:math.GT/0607509 , pages 331–350, 2011. [Mj14] M. Mj. Cannon-Thurston Maps for Surface Groups. Ann. of Math. 179 (1) , pages 1–80, 2014. [MP11] M. Mj and A. Pal. Relative Hyperbolicity, Trees

of Spaces and C annon-Thurston Maps. Geom. Dedicata 151, arXiv:0708.3578 , pages 59–78, 2011. [PH92] R. Penner and J. Harer. Combinatorics of train tracks. Ann. Math. Studies 125, Prince- ton University Press , 1992. [Sha91] P. B. Shalen. Dendrology and its applications. Group Theory from a Geometrical View- point (E. Ghys, A. Haefliger, A. Verjovsky eds.) , pages 543–616, 1991. [Thu80] W. P. Thurston. The Geometry and Topology of 3-Manifold s. Princeton University Notes , 1980.
Page 22
22 MAHAN MJ WITH SHUBHABRATA DAS RKM Vivekananda University, Belur Math, WB-711 202, India

E-mail address mahan.mj@gmail.com; mahan@rkmvu.ac.in E-mail address shubhabrata.gt@gmail.com