Limits of limit sets II: Geometrically Infinite Groups
LLIMITS OF LIMIT SETS II:GEOMETRICALLY INFINITE GROUPS
MAHAN MJ AND CAROLINE SERIES
Abstract.
We show that for a strongly convergent sequence of purely loxodromic finitelygenerated Kleinian groups with incompressible ends, Cannon-Thurston maps, viewed as mapsfrom a fixed base limit set to the Riemann sphere, converge uniformly. For algebraically con-vergent sequences we show that there exist examples where even pointwise convergence ofCannon-Thurston maps fails.
MSC classification: 30F40; 57M50Keywords: Kleinian group, limit set, Cannon-Thurston map, geometrically infinitegroup
Contents
1. Introduction 22. Background 52.1. Kleinian groups 52.2. Balls and geodesics 52.3. The Cayley graph 62.4. Algebraic and Geometric Convergence 62.5. Scott cores 72.6. Relative Hyperbolicity and Electric Geometry 93. Cannon-Thurston Maps and Convergence Criteria 103.1. Criteria for convergence 114. Strong Convergence 144.1. The bounded geometry case 144.2. Unbounded geometry 165. Algebraic limits and non-convergence of limit points 235.1. Brock’s Examples 235.2. Absence of Uniform Convergence 285.3. Pointwise non-convergence 30
Date : October 8, 2018.Research of first author partially supported by CEFIPRA project 4301-1. a r X i v : . [ m a t h . G T ] J un MAHAN MJ AND CAROLINE SERIES
Introduction
Given an isomorphism between Kleinian groups, a Cannon-Thurston map is a continuousequivariant map between their limit sets. It is by no means obvious that such a map alwaysexists, however as the culmination of a long series of developments, it was shown in [37] thatgiven a weakly type preserving (see below) isomorphism between any geometrically finite groupΓ and any Kleinian group G , a Cannon-Thurston map always exists.This paper is the second of two dealing with convergence of Cannon-Thurston or CT -maps,considered as a sequence of continuous maps from the limit set of a fixed geometrically finitegroup to the sphere. The main questions addressed in both papers are:(1) Does strong convergence of finitely generated Kleinian groups imply uniform conver-gence of CT -maps?(2) Does algebraic convergence of finitely generated Kleinian groups imply pointwise con-vergence of CT -maps?In the first paper [38] we dealt with the geometrically finite case by showing that both questionshave a positive answer for a sequence of geometrically finite groups converging to a geometricallyfinite limit, provided that, in case (2), the geometric limit is also geometrically finite. Asobserved in [38], it is easy to see that if the groups converge algebraically but not strongly, thenuniform convergence necessarily fails.In the present paper we study the situation in which the limit group is geometricallyinfinite. We show that, in the absence of parabolics and with incompressible ends, the answerto (1) is always positive, but, in what is the most unexpected outcome of our investigations,we provide a counter example to (2) by exhibiting a sequence of geometrically finite groupsconverging algebraically but not strongly, for which the corresponding CT -maps fail to convergepointwise at a countable set of points. The class of limit groups in question are Brock’s partiallydegenerate examples [9], described in more detail below. Thus our second main result answersin the negative the second part of Thurston’s Problem 14 in his seminal paper [43]. In theseexamples, both the algebraic and geometric limits of the G n are geometrically infinite. Wedo not know whether there exist examples of non-convergence in which the algebraic limit isgeometrically finite but the geometric limit is not.Recall that an isomorphism ρ : Γ → G between Kleinian groups is strictly type preserving if ρ ( γ ) ∈ G is parabolic if and only if γ ∈ Γ is also parabolic; it is weakly type preserving if the
IMITS OF LIMIT SETS II 3 image of any parabolic element is parabolic. Since CT -maps preserve fixed points, is easy tosee that a necessary criterion for the existence of a CT -map ˆ i : Λ Γ → Λ G between limit sets isthat ρ be weakly type preserving.Our first main result, largely answering question (1), is: Theorem A.
Let Γ be a geometrically finite Kleinian group without parabolics, which does notsplit as a free product. Let ρ n : Γ → G n be a sequence of strictly type preserving isomorphisms togeometrically finite Kleinian groups G n , which converge strongly to a totally degenerate purelyloxodromic Kleinian group G ∞ = ρ ∞ (Γ) . Then the sequence of CT -maps ˆ i n : Λ Γ → Λ G n converges uniformly to ˆ i ∞ : Λ Γ → Λ G ∞ . This result was proved by Miyachi [32] in the case in which Γ is a surface group withoutparabolics and the injectivity radius is uniformly bounded below along the whole sequence.The condition that Γ does not split as a free product is equivalent to requiring that all ends ofthe manifold H / Γ are incompressible, see [5].Theorem A of [38], which is essentially the above result in the geometrically finite case, doesnot have any of the restrictions (absence of parabolics, not splitting as a free product, strictlytype preserving, totally degenerate) imposed above. We introduce these restrictions largelybecause of technical issues concerning the model for the ends of the limit manifold H /G ∞ .With a bit more work, similar techniques to those used here can be used to prove the theoremin the general case, see [40].If the convergence is algebraic but not strong, then uniform convergence necessarily fails.This is an immediate consequence of Evans’ theorem [17, 18] that the limit sets Λ G n converge inthe Hausdorff metric to the limit set of the geometric limit in the Hausdorff metric, see [38] forfurther discussion. As far as we know, the question of pointwise convergence in this situationhas not previously been addressed. In answer to question (2) we have: Theorem B.
Let Γ be a Fuchsian group for which H / Γ is a closed surface of genus at least . Then there exists a Kleinian group G ∞ , together with an isomorphism ρ ∞ : Γ → G ∞ = ρ ∞ (Γ) , and a sequence of representations ρ n : Γ → G n to geometrically finite groups convergingalgebraically to G ∞ , such that the sequence of CT-maps ˆ i n : Λ Γ → Λ G n fails to convergepointwise to ˆ i ∞ at a countable set of points in Λ Γ . Implicit in the statement of Theorem A is the existence of the CT -map from Λ Γ to Λ G ∞ .This result has a long history which we do not intend to repeat in detail here. The mostgeneral result, in which G = G ∞ is an arbitrary torsion free non-elementary Kleinian group,can be found in [37]. The restricted case in which Γ is a surface group and G is singly or MAHAN MJ AND CAROLINE SERIES doubly degenerate, is the main result of [39], see Section 4 below. The original seminal case inwhich M = H /G is the cyclic cover of a 3-manifold fibering over the circle with fibers a closedsurface is of course due to Cannon and Thurston [14]; this was extended to the case in which G is a surface group with a lower bound on the lengths of loxodromics in [7] and [31], or moregenerally when M is an arbitrary hyperbolic manifold with incompressible boundary in [34].The older history in the case in which G is geometrically finite is discussed in [38].In [38] we introduced general criteria for uniform and pointwise convergence of CT -maps,called UEPP and EPP respectively. These compare the geometry of the obvious embedding ofthe Cayley graph of the base group Γ into H , to the corresponding embeddings for the groups G n , G ∞ . The main work in this paper consists in verifying that these criteria hold (in the caseof Theorem A) or understanding why they do not (in the case of Theorem B). After slightlyreformulating the condition UEPP, we see that for the case of strictly type preserving maps ofsurface groups, the needed condition has essentially already been proved in [39]. In order toexplain this, we give in Section 4.2 a brief outline of the relevant parts of the arguments in [39].For the benefit of readers who have not gone through all of this previous work, which in turndepends heavily on the Minsky model of degenerate Kleinian groups, we preface this by brieflysketching in Section 4.1 how the argument goes in the case of groups of bounded geometry,thus reproving Miyachi’s theorem [32]. Our proof in this case follows easily using the methodexplained in [31] and [36] and is independent of [39].Let R be a surface with boundary and Λ a lamination on R . A complementary region in R \ Λ is called a crown domain if it contains a component of ∂R . The counter example inTheorem B arises from Brock’s examples of a sequence of quasi-Fuchsian groups G n convergingalgebraically but not strongly to a partially degenerate group G ∞ . More precisely, we provethe following, which immediately implies Theorem B: Theorem C.
Fix a closed hyperbolizable surface S together with a separating simple closedcurve σ , dividing S into two pieces L and R . Let α denote an automorphism of S such that α | L is the identity and α | R = χ is a pseudo-Anosov diffeomorphism of R fixing the boundary σ . Let X be a hyperbolic structure on S and let G n be the quasi-Fuchsian group given by thesimultaneous uniformization of ( α n ( X ) , X ) . Let G ∞ denote the algebraic limit of the sequence G n , suitably normalized by a basepoint in the lift of the lower boundary X . Let ˆ i n : Λ G → Λ G n , n ∈ N ∪ ∞ , be the corresponding CT -maps and let ξ ∈ Λ G . Then ˆ i n ( ξ ) converges to ˆ i ∞ ( ξ ) ifand only if ξ is not the endpoint of the lift to H of a boundary leaf, other than σ , of the crowndomain of the unstable lamination of χ , viewed as a lamination on the surface R . IMITS OF LIMIT SETS II 5
The outline of the paper is as follows. In Section 2 we set up background and notation,in particular reviewing briefly what we need from the theory of hyperbolic spaces and electricgeometry in 2.6. These techniques are central in [39], and are also used here in the discussionof Theorem C.In Section 3 we recall results from [38] on Cannon-Thurston maps, in particular we explainour convergence criterion UEPP. In Section 4 we prove Theorem A. As discussed above, we firstgive a brief discussion of a proof in the case of bounded geometry, that is, when the injectivityradius of all manifolds in the sequence is uniformly bounded below. This is essentially Miyachi’stheorem referred to above. We then turn to the general situation, outlining as we go therelevant steps in the proof for a single CT -map as in [39]. Finally in Section 5 we explain thecounter examples to pointwise convergence, explaining the Brock examples and then provingTheorem C. Acknowledgments:
This work was done in part while the first author was visiting Universit´eParis-Sud XI under the Indo-French collaborative programme ARCUS. He gratefully acknowl-edges their support and hospitality. 2.
Background
Kleinian groups. A Kleinian group G is a discrete subgroup of P SL ( C ). As such itacts as a properly discontinuous group of isometries of hyperbolic 3-space H , whose boundarywe identify with the Riemann sphere ˆ C = C ∪ ∞ . As in [38], all groups in this paper will befinitely generated and torsion free, so that M = H /G is a hyperbolic 3-manifold. The limitset Λ G ⊂ ˆ C is the set of accumulation points of any G -orbit.A Kleinian group is geometrically finite if it has a fundamental polyhedron in H withfinitely many faces; a group which is not geometrically finite is also called degenerate . Thepoint of this paper is to investigate the extension of [38] to the degenerate case. We say agroup is totally degenerate if it is not geometrically finite and Λ G = ˆ C . The structure ofdegenerate groups has recently been elucidated by the work of Minsky et al. [29, 10] on theending lamination theorem and the tameness theorem of Agol [1] and Calegari and Gabai [11].This paper rests heavily on these results.A Kleinian group G is a surface group , if there is a hyperbolic surface S , together with adiscrete faithful representation ρ : π ( S ) → G . The corresponding manifold H /G is homeo-morphic to S × R , see [5]. It is singly or doubly degenerate according as one or both of its endsare geometrically infinite with filling ending laminations.2.2. Balls and geodesics.
We will be working in hyperbolic space H n for n = 2 ,
3. We denotethe hyperbolic metric on H n by d H or occasionally d H n ; sometimes we explicitly use the ball MAHAN MJ AND CAROLINE SERIES model B with centre O and denote by d E the Euclidean metric on B ∪ ˆ C . For P ∈ H n , write B ( P ; R ), or when needed B H ( P ; R ) or even B H n ( P ; R ), for the hyperbolic ball centre P andradius R . Let β be a path in H n with endpoints X, Y . We write [ β ] or [ X, Y ] for the H n -geodesicfrom X to Y .2.3. The Cayley graph.
Let G be a finitely generated Kleinian group with generating set G ∗ = { e , . . . , e k } . We assume throughout that G ∗ is symmetric, in the sense that g ∈ G ∗ ifand only if g − ∈ G ∗ for any g ∈ G . The Cayley graph G G of G is the graph whose vertices areelements g ∈ G and which has an edge between g, g (cid:48) whenever g − g (cid:48) ∈ G ∗ . The graph metric d G is defined as the edge length of the shortest path between vertices so that d G (1 , e i ) = 1 forall i , where 1 is the unit element of G . Let | g | denote the word length of g ∈ G with respect to G ∗ , so that | g | = d G (1 , g ). For X ∈ G G , we denote by B G ( X ; R ) ⊂ G G the d G -ball centre X and radius R .Choose a basepoint O G ∈ H which is not a fixed point of any element of G . One mayif desired assume the basepoint is the centre O of the ball model B as above. For simplicity,we do this throughout the paper unless indicated otherwise. Then G G is immersed in H bythe map j G which sends g ∈ G to j G ( g ) = g · O , and which sends the edge joining g, g (cid:48) to the H -geodesic joining j G ( g ) , j G ( g (cid:48) ). In particular, j G (1) = O . Note that using the ball model of H , the limit set Λ G may be regarded as the completion of j G ( G G ) in the Euclidean metric d E on B ∪ ˆ C .2.4. Algebraic and Geometric Convergence.
Let Γ be a geometrically finite Kleiniangroup. A sequence of group isomorphisms ρ n : Γ → P SL ( C ) , n = 1 , . . . is said to con-verge to the representation ρ ∞ : Γ → P SL ( C ) algebraically if for each g ∈ Γ, ρ n ( g ) → ρ ∞ ( g )as elements of P SL ( C ). The representations converge geometrically if ( G n = ρ n ( G )) con-verges as a sequence of closed subsets of P SL ( C ) to G g ⊂ P SL ( C ). Then G g is a Kleiniangroup called the geometric limit of ( G n ). The sequence ( ρ n ) converges strongly to ρ ∞ ( G )if ρ ∞ ( G ) = G g and the convergence is both geometric and algebraic. If a sequence of groupsconverge algebraically, they have a geometrically convergent subsequence, see [24] Theorem4.4.3.Alternatively, following Thurston [42], see also for example [12] Chapter 3, we say thata sequence of manifolds with base-frames ( M n , ω n ) converges geometrically (or in the C ∞ -Gromov-Hausdorff topology) to a manifold with base-frame ( M ∞ , ω ∞ ) if for each compactsubmanifold C ⊂ M ∞ containing the base-frame ω ∞ , there are smooth embeddings ψ n : C → M n (for all sufficiently large n ) which map base-frame to base-frame and such that ψ n convergesto an isometry in the C ∞ -topology. Kleinian groups G n are said to converge geometrically IMITS OF LIMIT SETS II 7 to G ∞ if the corresponding framed manifolds ( M n = H /G n , ω n ) converge geometrically to( M ∞ = H /G ∞ , ω ∞ ), where the base-frames ω n , ω ∞ are all the projection of a fixed base-framein H . The sequence (( M n , ω n )) converges strongly to ( M ∞ , ω ) if the convergence is geometricand in addition the convergence of ( ρ n ) to ρ ∞ is algebraic. We remark that changing thebasepoints in the above discussion may result in a different geometric limit.The relation between these definitions is the following. Fix once and for all a standardbase frame Ω in H , with basepoint at the origin O in the ball model of hyperbolic 3-space.Given a framed manifold ( M, ω ), there is a unique developing map ( (cid:102)
M , ˜ ω ) → H (where (cid:102) M isthe universal cover of M ) which sends a fixed lift ˜ ω of ω to Ω ∈ H . The induced holonomyhomomorphism sends π ( M, o ) to a discrete torsion free subgroup of SL (2 , C ), where o ∈ M is the basepoint of ω . By for example [12] Theorem 3.2.9, this map is a homeomorphism withappropriate topologies, so that convergence of manifolds in the sense of Thurston is equivalentto geometric convergence in the first sense defined above, see for example [24] Chapter 4 or [23]Theorem 8.11. In particular, the map ψ n : C → M n is the projection to the quotient manifoldsof a bi-Lipschitz embedding ˜ ψ n : B ( O ; R ) → H where B ( O ; R ) ⊂ H is a large ball whoseprojection to H /G ∞ contains C , see for example [3] Lemma 9.6.2.5. Scott cores.
Recall that a Scott core of a 3-manifold V is a compact connected 3-submanifold K V such that the inclusion K V (cid:44) → V induces an isomorphism on fundamentalgroups. The Scott core is unique up to isotopy [25]. Note that in general, the Scott core maybe much smaller than the convex core, even when the group is convex cocompact. We shallneed the following relationship between the Scott core and the ends of V . Lemma 2.1 ([5] Proposition 1.3, [23] Theorem 4.126) . Let K V be a Scott core of a -manifold V . There is a bijective correspondence between the ends of V and boundary components of V \ K V . Hence each component of ∂K V bounds a non-compact component of V \ K V and eachof these components is an end of V . Let ρ n : Γ → P SL ( C ) be a sequence of representations converging strongly to ρ ∞ . Fixingthe base-frames as the projections ω n of the frame Ω in H to M n = H /G n , n ∈ N ∪∞ , we obtaina corresponding sequence of framed hyperbolic manifolds ( M n , ω n ) converging geometrically to M ∞ = H /G ∞ . Set N = H / Γ and let K N be a Scott core of N , chosen such that the baseframe ω N for N has basepoint o N ∈ K N . The representations ρ n induce homotopy equivalences φ n : K N → M n . We can lift φ n to maps ˜ φ n : (cid:101) K N → (cid:102) M n for n ∈ N ∪ ∞ and note that by ourchoices that ˜ φ n (˜ ω N ) converges to ˜ φ ∞ (˜ ω N ). MAHAN MJ AND CAROLINE SERIES
In general the homotopy equivalences φ n may not be homeomorphisms. However in thesituation of strong convergence, the proof of [13] Proposition 3.3 or [3] Lemma 9.7, (see alsoLemma 3.6 and the first part of the proof of Theorem A in [4]) gives: Lemma 2.2.
Let Γ be a geometrically finite group and let ρ n be a sequence of discrete faith-ful representations of Γ converging strongly to ρ ∞ . Let K = K M ∞ be a compact core for M ∞ = H /ρ ∞ (Γ) . Let ψ n : K → M n be the bi-Lipschitz embeddings coming from the geo-metric convergence, inducing maps ( ψ n ) ∗ : ρ ∞ (Γ) → ρ n (Γ) . Then for all large enough n , ( ψ n ) ∗ = ρ n ◦ ρ − ∞ and ψ n ( K ) is a compact core for M n = H /ρ n (Γ) . We remark that the hypotheses in [13] and [3] that all groups be purely hyperbolic, orindeed that the convergence be strictly type preserving, are not needed for this lemma. Alsonote that as remarked above, in general ψ n ( K ) may be much smaller than the convex coreof M n . If the convergence is not strong, Lemma 2.2 may fail even when the limit group isgeometrically finite, as is shown by the well known Anderson-Canary examples [2].Lemma 2.2 shows that, in the situation of strong convergence, we may take the homotopyequivalences φ n to be homeomorphisms between Scott cores of the relevant groups. It alsoallows us to identify the ends of M ∞ with the ends of the approximating groups. We have: Corollary 2.3.
Let Γ be a geometrically finite group and let ρ n be a sequence of discrete faithfulrepresentations of Γ converging strongly to ρ ∞ . Then, up to replacing Γ by the group G n forsome n ∈ N , we can pick a Scott core K of H / Γ such that there are bi-Lipschitz embeddings φ n : K → H /G n which induce ρ n , and such that φ n ( K ) is a Scott core of M n for n ∈ N ∪ ∞ .Moreover suppose that E is an end of M ∞ and U is the component of M ∞ \ φ ∞ ( K ) which isa neighborhood of E . Let F = ∂U and let U n be the component of M n \ φ n ( K ) bounded by φ n φ − ∞ ( F ) . Then U n is a neighborhood of an end E n of M n and we say that U n corresponds to E .Proof. Set K = φ ∞ ( K ). Take n large enough that the conclusion of Lemma 2.2 applies.Replacing N = H / Γ by M n , Γ by G n , and ρ n by ρ (cid:48) n = ρ n ρ − n , we have a sequence ofrepresentations as before. The core of M n can be taken to be ψ n ( K ). Noting that ρ (cid:48) n isinduced by ψ n ψ − n : ψ n ( K ) → M n , we can replace the homotopy equivalences φ n : K N → M n by the homeomorphisms ψ n ψ − n : ψ n ( K ) → ψ n ( K ) between the cores of M n and M n , n > n .These converge to the homeomorphism ψ − n : ψ n ( K ) → K . In other words, we may as wellassume that the homotopy equivalences φ n are actually embeddings of the core K N of N into M n , M ∞ .The idea of the final statement follows [13] §
8. That U n is a neighborhood of an end E n of M n follows from Lemma 2.1. (cid:3) IMITS OF LIMIT SETS II 9
Relative Hyperbolicity and Electric Geometry.
We summarize the facts we need onrelative hyperbolicity and electric geometry. For further details, we refer the reader to [20, 6]see also [35] Section 3.Let (
X, d ) be a δ -hyperbolic metric space, and let H be a collection of pairwise disjointsubsets. To electrocute H means to construct an auxiliary metric space ( X el , d el ) in which thesets in H effectively have zero diameter, although for technical reasons it is preferable theyhave diameter 1 (or 2). Precisely, let X el = X (cid:83) H ∈H ( H × [0 , H × { } identified to H ⊂ X . We define the electric (pseudo)-metric d el on X el as follows. First equip H × [0 ,
1] withthe product metric and then modify this to a pseudo-metric by quotienting so that H × { } isequipped with the zero metric. The metric on H × [0 ,
1] is the path metric induced by horizontaland vertical paths. This means that in the space ( X el , d el ), any two points in H are at distanceat most 2. Definition 2.4. [20, 6]
Let X be a metric space and H be a collection of mutually disjointsubsets. If X el is also a hyperbolic metric space, then X is said to be weakly hyperbolic relativeto the collection H . The collection H is said to be uniformly separated if there exists C > d ( H i , H j ) ≥ C for all H i (cid:54) = H j ∈ H . It is uniformly quasi-convex if there exists C > H ∈ H and for any points x, x (cid:48) ∈ H , any geodesic joining them lies within the C -neighborhoodof H . It is mutually cobounded if there exists C > H i (cid:54) = H j ∈ H , π i ( H j ) hasdiameter less than C , where π i denotes a nearest point projection of X onto H i . Lemma 2.5 ([6], [20] Proposition 4.6) . Let X be a hyperbolic metric space and H a collectionof uniformly quasi-convex mutually cobounded uniformly separated subsets. Then X is weaklyhyperbolic relative to the collection H . A typical example is when X is hyperbolic space H and H is the collection of lifts to H of the thin parts of a hyperbolic 3-manifold.Following Farb, we need to understand some finer details of the relationship betweengeodesics in X and in X el . Recall that K-quasi-geodesic in a metric space Y is a K -quasi-isometric embedding of an interval into Y , that is, a map f : [ a, b ] → Y such that1 K | t − t | − K ≤ d Y ( f ( t ) , f ( t )) ≤ K | t − t | + K for all t , t ∈ [ a, b ]. A quasi-geodesic is a path in Y which is a K -quasi-geodesic for some K >
0. If X, H gives rise to an electric space ( X el , d el ), then an electric (quasi)-geodesic in X is a path in X which is a (quasi)-geodesic for the electric metric d el . We say that a path does not backtrack if it does not re-enter any H ∈ H after leavingit. Suppose that λ is an electric quasi-geodesic in ( X el , d el ) without backtracking and withendpoints a, b ∈ X \ H . Keeping the endpoints a, b fixed, replace each maximal subsegment of λ lying within some H ∈ H by a hyperbolic X -geodesic with the same endpoints. The resultingconnected path is called an electro-ambient quasi-geodesic in X , see Figure 1. The main resultwe need is: Figure 1.
An electro-ambient quasi-geodesic.
Lemma 2.6 ([35] Lemma 3.7) . Let X be a hyperbolic metric space and let H be a collectionof mutually cobounded uniformly separated uniformly quasi-convex sets. Let γ be an electro-ambient quasi-geodesic with endpoints a, b ∈ X \ H . Then γ is a quasi-geodesic in X and lieswithin bounded distance of any X -geodesic with the same endpoints. In the case in which X is hyperbolic space H and H is the collection of lifts to H of thethin parts of a hyperbolic 3-manifold, the proof of Lemma 2.6 is straightforward and was donefrom first principles in [38] Lemmas 7.15, 7.16. Remark 2.7.
One can introduce a further property of a metric space X being strongly hyper-bolic relative to a collection H , see [35] Section 3 and also [6]. This condition concerns howpaths penetrate the sets in H . If X is itself a δ -hyperbolic space, then the conditions that thesets in H be mutually cobounded, uniformly separated and uniformly quasi-convex are enoughto imply that X is strongly hyperbolic relative to H , see [20] § §
3. Since all that weneeded here is the result of Lemma 2.6, we do not digress to give the precise definition here.3.
Cannon-Thurston Maps and Convergence Criteria
Let Γ be a Kleinian group and let ρ : Γ → P SL ( C ) with ρ (Γ) = G . Let Λ Γ , Λ G be thecorresponding limit sets. A Cannon-Thurston or CT -map is an equivariant continuous mapˆ i : Λ Γ → Λ G , that is, a map such thatˆ i ( g · ξ ) = ρ ( g )ˆ i ( ξ ) for all g ∈ Γ , ξ ∈ Λ Γ . IMITS OF LIMIT SETS II 11
Recall from Section 2.3 the natural embedding j Γ of the Cayley graph of G Γ into H . The CT -map ˆ i = ˆ i ( ρ ) can also be defined as the continuous extension to Λ Γ ⊂ ∂ H of the obviousmap i : j Γ ( G Γ) → H defined by i ( j Γ ( g )) = ρ ( g ) · O .Suppose alternatively that N is a geometrically finite manifold homotopy equivalent toanother hyperbolic manifold M and let φ : K N → M be a homotopy equivalence between aScott core K N of N and M . It is easy to see that ˆ i is an extension to ∂ H of any lifting (cid:101) φ : (cid:101) K N → (cid:102) M , since any fixed orbit of the action of G on ˜ φ ( ˜ K N ) can serve as a substitute forthe orbit of the basepoint O . Notice however that the careful discussion in 2.4 is needed to fixbasepoints if we are dealing with a sequences of groups.In [38] we reproved Floyd’s results [21] on the existence of CT -maps for geometrically finitegroups: Theorem 3.1 ([38] Theorem 4.2) . Let Γ , G be finitely generated geometrically finite groups andlet φ : Γ → G be weakly type preserving isomorphism. Then the CT -map ˆ i : Λ Γ → Λ G exists. Note that the examples in [2] show that there exist geometrically finite non-cusped manifoldswhich are homotopic but not homeomorphic, see also Lemma 2.2. We deduce from the abovethat the CT -map between their limit sets nonetheless exists and is a homeomorphism.3.1. Criteria for convergence.
We now collect some criteria for the existence and conver-gence of CT -maps from [38]. Theorem 3.2 ([38] Theorem 4.1) . Let ρ : Γ → G be a weakly type preserving isomorphismof finitely generated Kleinian groups and suppose that Γ is geometrically finite. The CT -map Λ Γ → Λ G exists if and only if there exists a function f : N → N , such that f ( N ) → ∞ as N → ∞ , and such that whenever λ is a d Γ -geodesic segment lying outside B Γ (1; N ) in G Γ , the H -geodesic joining the endpoints of i ( j Γ ( λ )) lies outside B H ( O ; f ( N )) in H . In [39], the criterion was used in an alternative form which involves geodesics in H for thedomain group Γ also. Recall that a geometrically finite group is convex cocompact if its convexcore is compact. In this case we can take the Scott core to be the convex core. Theorem 3.3.
Let ρ : Γ → G be a strictly type preserving isomorphism of finitely generatedKleinian groups, and suppose that Γ is convex cocompact. Suppose that K N is the convex core of N = H / Γ and that φ : K N → H /G is a homotopy equivalence. Then the CT -map Λ Γ → Λ G exists if and only if there exists a non-negative function f : N → N , such that f ( M ) → ∞ as M → ∞ , and such that whenever λ is a H -geodesic segment lying outside B ( O ; M ) , the H -geodesic [ ˜ φ ( λ )] lies outside B ( ˜ φ ( O ); f ( M )) , where ˜ φ is a fixed lift of φ to the obvious mapfrom (cid:103) K N (the convex hull of Λ Γ ) to H . Note that to make a sensible statement here we need to insist that, in addition to beingcompact, the core K N of N should also be convex. If Γ is Fuchsian group corresponding to aclosed surface, the main case in [39], then this is not an issue.If Γ is convex cocompact, then to compare geodesics in G Γ and H , one can use that any d Γ -geodesic in G Γ is a quasi-geodesic in H , see for example [38] Corollary 3.8. Moreover anygeodesic in H can be tracked at bounded distance by a path which is geodesic in G Γ (withbound depending only on Γ), see [38] Lemma 3.6. This shows the equivalence of the criteria inTheorems 3.2 and 3.3 in this case.
Remark 3.4.
If Γ is not convex cocompact then the comparison between geodesics in G Γ and H is not quite straightforward, since a geodesic path in G Γ can track round the boundary ofa horosphere. This can be dealt with but requires more care. In Theorem A of this paper, Γis always convex cocompact, and in any case, we shall mainly stick to the first version of thecriterion.Now let Γ be a fixed geometrically finite Kleinian group and suppose that ρ n : Γ → P SL ( C )is a sequence of discrete faithful weakly type preserving representations converging algebraicallyto ρ ∞ : Γ → P SL ( C ). Let G n = ρ n (Γ) , n ∈ N ∪ ∞ and write Λ n for Λ G n . To normalize, weembed all the Cayley graphs with the same base-point O = O G n for all n and set j n ( g ) = ρ n ( g ) · O, g ∈ G
Γ. Define i n : j Γ ( G Γ) → H by i n ( j Γ ( g )) (cid:55)→ ρ n ( g ) · O, g ∈ G
Γ, so that the CT -map ˆ i n : Λ Γ → Λ n is the continuous extension of i n to Λ Γ . Assuming they exist, we saythat the CT -maps ˆ i n : Λ Γ → Λ n converge uniformly (resp. pointwise) to ˆ i ∞ if they do so asmaps from Λ Γ to ˆ C . If λ is any d Γ -geodesic segment in G Γ with endpoints γ, γ (cid:48) ∈ Γ, then wealso write [ j ( λ )] for the H -geodesic [ j ( γ ) , j ( γ (cid:48) )].In [38] we introduced two properties UEP (Uniform Embedding of Points) and UEPP(Uniform Embedding of Pairs of Points) of the sequence ( ρ n ). The first was shown in [38] tobe equivalent to strong convergence, and the second is our criterion for uniform convergence of CT -maps. We summarise these definitions and results here. Definition 3.5.
Let ρ n : Γ → G n be a sequence of weakly type preserving isomorphisms ofKleinian groups. Then ( ρ n ) is said to satisfy UEP if there exists a non-negative function f : N → N , with f ( N ) → ∞ as N → ∞ , such that for all g ∈ Γ , d Γ (1 , g ) ≥ N implies d H ( ρ n ( g ) · O, O ) ≥ f ( N ) for all n ∈ N . Definition 3.6.
Let ρ n : Γ → G n be a sequence of weakly type preserving isomorphisms ofKleinian groups. Then ( ρ n ) satisfies UEPP if there exists a function f (cid:48) : N → N , such that f (cid:48) ( N ) → ∞ as N → ∞ , and such that whenever λ is a d Γ -geodesic segment lying outside B Γ (1; N ) in G Γ , the H -geodesic [ j n ( λ )] lies outside B H ( O ; f (cid:48) ( N )) for all n ∈ N . IMITS OF LIMIT SETS II 13
Proposition 3.7 ([38] Proposition 5.3) . Suppose that a sequence of discrete faithful weaklytype preserving representations ( ρ n : Γ → P SL ( C )) converges algebraically to ρ ∞ . Then ( ρ n ) converges strongly if and only if it satisfies UEP. Theorem 3.8 ([38] Theorem 5.7) . Let Γ be a geometrically finite Kleinian group and let ρ n :Γ → G n be weakly type preserving isomorphisms to Kleinian groups. Suppose that ρ n convergesalgebraically to a representation ρ ∞ . Then if ( ρ n ) satisfies UEPP, the CT -maps ˆ i n : Λ Γ → Λ n converge uniformly to ˆ i ∞ . If Γ is non-elementary, the converse also holds. Inverting the function f (cid:48) in the definition of UEPP gives a slight modification of the criterionin Theorem 3.8 convenient to our purposes. Note that in the statement which follows, we donot need to assume that f ( N ) → ∞ as N → ∞ . Corollary 3.9.
Let Γ be a geometrically finite Kleinian group and let ρ n : Γ → G n be weaklytype preserving isomorphisms to Kleinian groups. Suppose that ρ n converges algebraically toa representation ρ ∞ and that there exists a function f : N → N such that for any L ∈ N ,whenever λ is a d Γ -geodesic segment lying outside B Γ (1; f ( L )) in G Γ , the H -geodesic [ j n ( λ )] lies outside B H ( O ; L ) for all n ∈ N . Then the CT -maps ˆ i n : Λ Γ → Λ n converge uniformly to ˆ i ∞ .Proof. In view of Theorem 3.8, it is enough to see the given condition implies UEPP. We dothis by inverting the function f . Precisely, say L (cid:48) > L . If λ is a d Γ -geodesic segment lyingoutside B Γ (1; f ( L (cid:48) )) in G Γ, the H -geodesic [ j n ( λ )] lies outside B H ( O ; L (cid:48) ) and hence certainlyoutside B H ( O ; L ). Thus we may modify f if needed so that f ( L (cid:48) ) > f ( L ). Hence withoutloss of generality using an inductive argument we may assume that f is strictly increasing, sothat in particular f ( N ) → ∞ as n → ∞ .Now given N ∈ N define f (cid:48) ( N ) = L , where L is the unique positive integer such that N ∈ [ f ( L ) , f ( L + 1)). Note that with this definition, f (cid:48) ( N ) → ∞ as N → ∞ .Suppose given N and that f (cid:48) ( N ) = L . We are given that whenever λ is a d Γ -geodesicsegment lying outside B Γ (1; f ( L )) in G Γ, the H -geodesic [ j n ( λ )] lies outside B H ( O ; L ) for all n ∈ N . Thus whenever λ is a d Γ -geodesic segment lying outside B Γ (1; N ) in G Γ, it also liesoutside B Γ (1; f ( L )) and hence the H -geodesic [ j n ( λ )] lies outside B H ( O ; L ) = B H ( O ; f (cid:48) ( N ))for all n ∈ N . This is exactly the condition UEPP. (cid:3) Applying a similar inversion to Theorem 3.2 we obtain immediately:
Corollary 3.10.
Let Γ be a geometrically finite Kleinian group and let ρ : Γ → G be a weaklytype preserving isomorphism to a non-elementary Kleinian group G for which a CT -map exists. Then there exists a function f : N → N such that for any L ∈ N , whenever λ is a d Γ -geodesicsegment lying outside B Γ (1; f ( L )) in G Γ , the H -geodesic [ j ( λ ))] lies outside B H ( O ; L ) . Strong Convergence
In this section we prove Theorem A. To set the scene, we begin with a brief discussion inthe case of strong convergence and bounded geometry, that is, when the injectivity radii of allmanifolds in the sequence are uniformly bounded below. We then turn to the general situationof Theorem A, outlining as we go the relevant steps in the proof for a single CT -map as in [39].4.1. The bounded geometry case.
In this section, which is not necessary for the generalresult, we illustrate how to use the criteria of Section 3 to prove Theorem A in the case of asingly or doubly degenerate surface group with bounded geometry. This is essentially Miyachi’sresult [32]. The brief sketch below is a reprise of the first author’s original arguments in [31, 36].This sketch may also be useful to clarify the flow of the arguments from [39] which are the basisof our proof of Theorem A.Recall that if M is a hyperbolic 3-manifold, the injectivity radius r ( M ; x ) of M at x ∈ M is r/ r is the length of the shortest loop based at x . The manifold is said to have boundedgeometry if there exists a > r ( M ; x ) ≥ a for all x ∈ M ; in this case the injectivityradius of M is inf x ∈ M r ( M, x ). Theorem 4.1.
Let Γ be a Fuchsian group such that H / Γ is a closed hyperbolic surface. Let ρ n : Γ → G n be a sequence of strictly type-preserving isomorphisms to geometrically finite groups G n , which converge strongly to a singly or doubly degenerate surface group G ∞ = ρ ∞ (Γ) .Suppose moreover that the injectivity radii of M n are uniformly bounded below by some a > for n ∈ N ∪ ∞ . Then the sequence of CT -maps ˆ i n : Λ Γ → Λ G n converges uniformly to ˆ i ∞ : Λ Γ → Λ G ∞ .Proof. It is sufficient to show that the sequence ( ρ n ) satisfies UEPP. That is, we have to showthat there exists a function f (cid:48) : N → N such that f (cid:48) ( N ) → ∞ as N → ∞ and such thatwhenever λ is a d Γ -geodesic segment lying outside B Γ (1; N ) in G Γ, the H -geodesic [ j n ( λ )] liesoutside B H ( O ; f (cid:48) ( N )) for all n ∈ N .By [31] Theorem 4.7, the Cannon-Thurston map exists for the group G ∞ . Hence, byTheorem 3.2, there exists a non-negative function f : N → N , such that f ( N ) → ∞ as N → ∞ ,and such that whenever λ is a d Γ -geodesic segment lying outside B Γ (1; N ) in G Γ, the H -geodesicjoining the endpoints of i ∞ ( j Γ ( λ )) lies outside B H ( O G ; f ( N )) in H . What we have to do is toshow that the same function f works uniformly for the representations ρ n for all sufficientlylarge n . IMITS OF LIMIT SETS II 15
Finding the function f in [31] was based on the following construction. For simplicity,suppose that G ∞ is doubly degenerate; the singly degenerate case is similar. Let S be atopological surface homeomorphic to H / Γ. By results of Minsky [27, 28], one can pick asequence of pleated surface maps S → S n ⊂ M ∞ = H /G ∞ , n ∈ Z , such that the distancebetween S n , S n +1 is uniformly bounded above and below, and such that, with respect to theinduced hyperbolic metrics on the S n , there are uniformly bounded quasi-isometries S n → S n +1 .One deduces that the universal cover (cid:102) M ∞ of M ∞ is quasi-isometric to a ‘tree’ of Gromovhyperbolic metric spaces. This is a Gromov hyperbolic space X equipped with a map P ontoa simplicial tree T , which in the case of a doubly degenerate surface group can be taken to bethe tree whose vertices are the integers, with a unit length edge joining n to n + 1 , n ∈ Z . Theinverse image of each vertex is itself a Gromov hyperbolic space; these spaces are uniformlyproperly embedded into X . Moreover for each pair of adjacent vertices v, v (cid:48) , there is a quasi-isometry between the spaces P − ( v ) , P − ( v (cid:48) ), again assumed to have quasi-isometry constantsuniform over vertices v .In the present case, each space P − ( n ) is quasi-isometric to the universal cover of S n . Inparticular the map P : P − (0) → X should be thought of as the lift to universal covers of themap S → M ∞ . Since S = H / Γ is by assumption a closed surface, its universal cover isquasi-isometric to the Cayley graph of Γ. If we assume that the lift O ∈ H of the basepoint o ∈ S maps to the lifted basepoint O ∈ H = (cid:102) M ∞ , then we can replace a geodesic λ as in thestatement of the theorem with a geodesic in the hyperbolic space P − (0), see the commentsfollowing Theorem 3.3.We have to compare λ with the hyperbolic geodesic [ λ ] in (cid:102) M = H joining its endpoints.The key idea is to construct a ‘ladder’ L λ ⊂ X by flowing λ up the levels in X using the quasi-isometries between the levels, see [31] for details. By constructing a coarse Lipschitz projectionfrom X to L λ , it is shown that L λ is uniformly quasi-convex in X . That is, there exists k > X -geodesic with ends in L λ lies within distance k of L λ . A short argument givenin the proof of [31] Theorem 3.10 shows that this is sufficient to construct the required function f for the manifold M .This function f depends on the quasi-isometry between (cid:102) M and X , and it is not hard tosee by inspection that the constants depend only on the injectivity radius of M . The constant k has a similar dependence. Thus if we have a family of manifolds all of which have the samelower bound on injectivity radii, the same function f works simultaneously for all M n and thecriterion of Corollary 3.9 is satisfied. (cid:3) The problem with directly extending this proof to the situation of unbounded geometry,is that it requires a ladder and projection whose constants depend on the injectivity radius of the whole end. This is clearly not possible in the case of unbounded geometry. However wenote that the same methods can be extended to the case of a surface with punctures, see [36]Section 5.5. (Essentially, this uses similar arguments about crossing horoballs to those in [38].)This gives the following result which we use in the proof of Proposition 5.12 in Section 5.
Theorem 4.2.
Let Γ be a Fuchsian group such that H / Γ is a finite area hyperbolic surface.Let ρ n : Γ → G n be a sequence of strictly type-preserving isomorphisms to geometrically finitegroups G n , which converge strongly to a singly or doubly degenerate surface group G ∞ = ρ ∞ (Γ) .Suppose moreover that the injectivity radii of M n outside cusps are uniformly bounded belowfor n = 1 , , . . . , ∞ . Then the sequence of CT -maps ˆ i n : Λ Γ → Λ G n converges uniformly to ˆ i ∞ : Λ Γ → Λ G ∞ . Unbounded geometry.
Our proof of Theorem A is based on the method in [39], whicheffectively verifies the condition of Theorem 3.2 for a single group. To show that UEPP holdsfor a sequence converging strongly to such a limit, we need to examine the argument carefully tounderstand the dependence of the constants on the limit group. We simplify by explaining thefirst part of the proof in the case of surface groups, discussing extensions to general manifoldswith incompressible boundary later.The proof of Theorem A follows very closely that of the main result of [39], which can beroughly stated as:
Theorem 4.3 ([39] Theorem 7.1) . Cannon-Thurston maps exist for simply or doubly degeneratesurface Kleinian groups without cusps.
The actual result on which we base the proof of Theorem A is [39] Corollary 6.13, whichwe restate in an equivalent formulation as Lemma 4.7 below. To give more insight into whatis involved, we begin by sketching the relevant parts of the proof of Theorem 4.3. For this wefirst require a brief digression on split geometry as introduced in [39], which we do in 4.2.1. In4.2.2 we sketch the proof of Theorem 4.3. In 4.2.3 we explain the main technical result we use,Corollary 4.10. Finally in 4.2.4 we prove Theorem A.4.2.1.
Split geometry.
Let S be a hyperbolizable surface and let M be a manifold homeomorphicto S × I where I ⊂ R is an interval either finite or infinite. To say that M has split geometrymeans, roughly speaking, that it is made by gluing together a succession (finite or infinite) ofso-called split blocks . These are blocks each of which is homeomorphic to S × [0 , split subsurface S s of a hyperbolic surface S is a (possibly disconnected) proper subsurfacewith boundary, whose components are all essential and non-annular, and whose complement in IMITS OF LIMIT SETS II 17 S is a non-empty family of non-homotopic annuli which are k -neighborhoods of non-peripheralgeodesics on S . Moreover S s is required to have bounded geometry, in the sense that thereexists some universal (cid:15) > S s is of length (cid:15) .A split block B s ⊂ B = S × [0 ,
1] is a topological product S s × [0 ,
1] for some split subsurface S s of S , with the qualification that its upper and lower boundaries are only required to be splitsub-surfaces of S s . Split blocks are glued along their boundaries to build up model manifolds inthe spirit of [29]. A split component is a connected component of a split block, see [39] followingDefinition 4.11.Suppose M is a model manifold obtained by gluing finitely or infinitely many split blocksalong their boundaries. Section 4.3 in [39] introduces a so-called graph metric d graph on theuniversal cover (cid:102) M . (This metric is denoted d G in [39]; the notation here is used to distinguishit from the word metric on the Cayley graph.) Roughly speaking d graph is obtained by firstelectrocuting the lifts of Margulis tubes in each split block, and then by electrocuting the liftsof connected components of each split block. A (Gromov) hyperbolic manifold is said to be of split geometry , see [39] Definition 4.31, if each split component is quasi-convex (not necessarilyuniformly) in the hyperbolic metric on (cid:102) M and if in addition, the hyperbolic convex hull of theuniversal cover of any split component has uniformly bounded diameter in the graph metric d graph . This uniform bound is called the graph quasi-convexity constant and plays a crucial rolein the discussion.Given a singly or doubly degenerate hyperbolic manifold M whose fundamental group isa surface group, a large part of the work in [39] is to construct a model manifold M of splitgeometry whose universal cover (cid:102) M is bi-Lipschitz homeomorphic to the universal cover (cid:102) M of M . The model M is made by consistently gluing finitely or infinitely many split blocks B i sothat B i − is glued to B i along their common boundary split surface S i . The existence of thesequence of split level surfaces and split blocks exiting the end is a consequence of the Minskymodel [29] for a degenerate end. The detailed construction is intricate and involves a carefulselection of the split level surfaces using the Minsky hierarchy, see [39] especially § § Theorem 4.1 ([39] Theorem 4.32) . The hyperbolic manifold corresponding to any singly ordoubly degenerate surface group without accidental parabolics is bi-Lipschitz homeomorphic toa model manifold with split geometry.
The part of this result which asserts uniform graph quasi-convexity of the blocks is [39]Proposition 4.23. We note the point, key for our purposes here, that the graph quasi-convexityconstant is a combinatorial quantity which depends only on the topological convexity of the surface defining the end, and is thus also uniform across all degenerate ends of any hyperbolicmanifold defined by the same topological surface S .4.2.2. Rough sketch of Theorem 4.3.
Let S = H / Γ be a compact hyperbolic surface and let ρ : Γ → G be a type preserving isomorphism to a singly or doubly degenerate surface group G .The criterion used in [39] to prove the existence of a CT -map for ρ is that given in Theorem 3.2,but it will be convenient for our purposes to use the alternative formulation of Corollary 3.10.In view of Theorem 4.1, we may work either with M = H /G , or with a quasi-isometric modelmanifold of split geometry M . Lifting to universal covers, we obtain an identification of theuniversal cover (cid:102) M of M with H , and in particular we can identify a basepoint O ∈ H with apoint, also denoted O , in (cid:102) M .Here is the statement we need: Proposition 4.4.
Let G be a totally degenerate surface group corresponding to a strictly typepreserving representation ρ : π ( S ) → G where S is a closed surface as above, and let M be amodel manifold of split geometry corresponding to M = H /G . Let B ⊂ S × [0 , be a fixedbase block and let φ : S → M be the embedding which identifies S with S × { } ⊂ B . Fix abasepoint O ∈ (cid:101) S in the universal cover (cid:101) S = H . Denote by (cid:101) φ the lift of φ such that (cid:101) φ ( O ) is thebasepoint O ∈ H = (cid:102) M .Then for any L ∈ N , there exists f ( L ) ∈ N , such that whenever λ is a geodesic segment in ( (cid:101) S, d S ) lying outside an f ( L ) -ball around O ∈ (cid:101) S (where d S is the lifted hyperbolic metric on (cid:101) S ),the hyperbolic geodesic [ (cid:101) φ ( λ )] in (cid:102) M joining the endpoints of (cid:101) φ ( λ ) lies outside the L -ball around O ∈ (cid:102) M . Proposition 4.4 follows from [39] Lemma 6.12, restated as Lemma 4.7 below. Theorem 4.3follows immediately from Proposition 4.4 on applying Corollary 3.10. The condition that G ∞ be totally degenerate is introduced simply to avoid the annoyance of having to deal withgeometrically finite ends. Remark 4.5.
We refer to the discussion in Section 3 for the equivalence of the condition asstated here with the condition on geodesics in the Cayley graph G Γ. Remark 4.6.
Strictly speaking, since S is S with some Margulis tubes deleted, it cannot beidentified with S . For a precise statement we need to work instead with welded split blocks inwhich the ends of the Margulis tubes deleted in the split blocks B i are reglued. Gluing thewelded split blocks along their boundaries gives a welded model manifold M wel homeomorphicto S × R or S × [0 , ∞ ) according as M is doubly or singly degenerate, see [39] § IMITS OF LIMIT SETS II 19
The essence of Proposition 4.4 is contained in Lemma 4.7 below, whose proof occupies § § λ (or more precisely,the image (cid:101) φ ( λ ) of λ in (cid:102) M ) to construct a ‘ladder’ L λ by ‘flowing up’ through the blocks of (cid:102) M . By constructing a coarse retraction onto L λ , it is shown ([39] Corollary 5.8) to be quasi-convex in the graph metric d graph . The constants here are independent not only of λ , butalso of the particular model M , depending in fact only on the topological type of the surface S . Starting from λ , this allows one to construct an ‘admissible’ path joining the endpointsof λ which follows the levels L λ and which, after a controlled sequence of alterations using asuccession of different metrics, is ultimately modified into an electro-ambient quasi-geodesic in( (cid:102) M , d CH ) joining the endpoints of λ . Here d CH is the metric on (cid:102) M obtained by electrocutingthe hyperbolic convex hulls CH ( (cid:101) K ) of the extended split components (cid:101) K of (cid:102) M . (Recall that anelectro-ambient quasi-geodesic in ( (cid:102) M , d CH ) is an electric quasi-geodesic whose intersection witheach electrocuted component is geodesic in the hyperbolic (or model) metric, see Section 2.6.)This finally leads to the following key statement, which we once again formulate in the modifiedform appropriate to Corollary 3.10: Lemma 4.7 ([39] Lemma 6.12) . Let M be a model manifold of split geometry and withoutcusps for a fixed compact hyperbolizable surface S . Let B ⊂ S × [0 , be a fixed base block andlet φ : S → M be an embedding which identifies S with S × { } ⊂ B . Let o ∈ S ⊂ M be abasepoint and fix a lift O in the universal cover (cid:101) S = H . Let φ be the lift (cid:101) φ : (cid:101) S → (cid:102) M = H sothat (cid:101) φ ( O ) = O as in Proposition 4.4.Then for any L > , there exists f ( L ) > , such that for any geodesic segment λ in ( (cid:101) S, d S ) lying outside B H ( O, f ( L )) ⊂ (cid:101) S , there is an electro-ambient quasi-geodesic γ in ( (cid:102) M , d CH ) joiningthe endpoints of (cid:101) φ ( λ ) which lies outside B H ( O, L ) ⊂ (cid:102) M . To get from Lemma 4.7 to the statement of Proposition 4.4, we have now to only to showthat one can replace electro-ambient quasi-geodesic γ in ( (cid:102) M , d CH ) by the hyperbolic geodesic[ i ( λ )] joining the endpoints of λ . This can be done in virtue of Lemma 2.6, see also [39] Lemma2.5 and the short argument in the proof of [39] Theorem 7.1.4.2.3. Proof of Theorem A: preliminaries.
We want to adapt the above discussion to the sit-uation of Theorem A, namely a sequence of geometrically finite purely loxodromic groups G n converging strongly to a degenerate group G ∞ .First, suppose that we have a single type preserving discrete faithful representation ρ : Γ → G where Γ is geometrically finite purely loxodromic group and G is a totally degenerate Kleiniangroup with incompressible ends. As discussed in § simply degenerate end of any hyperbolic manifold with incompressible ends. Thus one caneasily modify Lemma 4.7 to apply in this more general situation.The essence of the proof of Theorem A is therefore to understand the dependence of theconstants involved on the manifold M = H /G . The manifold, and hence the ladder L λ (reallyone ladder for each end of M ), may not have bounded geometry, since its construction dependson the whole of each infinite end of M . However as already noted, the Lipschitz constant forthe coarse projection to L λ in the graph metric d graph depends only on the topological type ofthe surfaces defining the ends and is thus uniform over any approximating sequence ρ n .Next, we need to look into the effect on the constants involved of the modifications needed toget from the d graph -electro-ambient geodesic joining the endpoints of (cid:101) φ ( λ ) as in [39] Section 6.1(called β e in [39]) to a d CH -electro-ambient geodesic with the same endpoints as in Lemma 4.7and hence to a hyperbolic geodesic as in Proposition 4.4. Suppose we have a bounded segment γ of such an electro-ambient geodesic which passes through a given finite collection of splitblocks. Close examination of the proof in [39] shows that the modifications made in the courseof replacing γ with the corresponding segment ˆ γ of the required hyperbolic geodesic, dependonly on the geometry of the hyperbolic convex hulls of the split components traversed by γ . The number N ( γ ) of convex hulls traversed is bounded uniformly in terms of the graphquasi-convexity constant. Thus the modifications made to γ , which control how much nearerˆ γ approaches the basepoint than γ , depend only the geometry of N ( γ ) blocks, where N ( γ )depends only on the topology of the ends of M .This leads to an equivalent reformulation of Lemma 4.7, which is essentially the same as [39]Corollary 6.13 in the context of a general hyperbolic manifold M without cusps. We begin withsome more notation. Suppose M has ends E , · · · , E r , and that each E k is homeomorphic to S k × [0 , ∞ ) for some closed hyperbolic surface S k , and such that each end is simply degenerate.Let K (cid:44) → M be a Scott core cutting off the ends E k , so that the boundary components of K are the surfaces S k × { } , i = 1 , . . . , r . Theorem 4.3 asserts that each end E = E k of M hassplit geometry and has a model made by consistently gluing split blocks B i = B ki , i ∈ N , sothat B i − is glued to B i along their common boundary split surface S i = S ki . For q ∈ N , let B ( q ) = K ∪ (cid:83) rk =0 (cid:83) qi =0 B ki be the manifold formed by gluing the core K to the first q blocksin each end as above. Replacing models by actual ends, we may assume that B ( q ) is quasi-isometrically embedded in M .We also change our formulation so as to be in accordance with the criterion in terms of G Γ, see the explanation in Section 3. Given an isomorphism ρ : Γ → G , we have an embedding j : G Γ → H , j ( γ ) = ρ ( γ ) · O of the Cayley graph of Γ into H . Recall also that if λ is any IMITS OF LIMIT SETS II 21 d Γ -geodesic segment in G Γ with endpoints γ, γ (cid:48) ∈ Γ, then we write j ( λ )] for the H -geodesic[ j ( γ ) , j ( γ (cid:48) )]. Remark 4.8.
We also have the map i : j Γ ( G Γ) → H defined by ij Γ ( γ ) = j ( γ ), whose extensionto Λ Γ is the CT -map ˆ i . Note that i is morally the same as the embedding (cid:101) φ : (cid:101) S → (cid:102) M ofProposition 4.4, and as long as S is closed, the map j Γ is a quasi-isometry so that i is alsomorally equivalent to j . Proposition 4.9.
Let Γ be a geometrically finite convex cocompact Kleinian group and let ρ : Γ → G be a strictly type preserving isomorphism. Suppose that M = H /G is a hyperbolic -manifold without cusps, with Scott core K and incompressible ends E , · · · , E r as above. Thenthere exists D ∈ N , depending only on the topology of the ends of M , with the following property.Suppose given q ∈ N and that the submanifold B ( q ) is defined as above. Then for all L > there exists f ( L ) > , depending only on the geometry of the submanifold B ( q + D ) , such thatif λ is any d Γ -geodesic segment in G Γ which lies outside B Γ (1 , f ( L )) , then [ j ( λ )] ∩ (cid:103) B ( q ) liesoutside B ( O, L ) , where (cid:103) B ( q ) is the lift to (cid:102) M of B ( q ) . Thus the proposition asserts that, independent of the geometry of M outside B ( q + D ), wecan control [ j ( λ )] inside (cid:103) B ( q ) with constants which depend only on the first q + D blocks ineach end, where D is a universal constant which depends only on the topological types of theends.We immediately deduce the following Corollary, which will be used in the proof of Theo-rem A. Corollary 4.10.
Let M, K , D be as in Proposition 4.9. Let q ∈ N . Then there exists f : N → N with the following property. Suppose that V is any hyperbolic manifold such that there is a bi-Lipschitz embedding β : B ( q + D ) → V , and such that the ends of V \ β ( B ( q + D )) have splitgeometry and correspond bijectively and homeomorphically to the ends of M \ B ( q + D ) . Let ˜ β : (cid:101) K → (cid:101) V be the lift of β to the universal covers. Suppose that L > and that λ is any d Γ -geodesic segment in G Γ which lies outside B Γ (1 , f ( L )) . Then [ (cid:101) β ( j ( λ ))] ∩ (cid:101) β ( (cid:101) B ( q )) lies outside B ( β ( O ) , L ) ⊂ (cid:101) V , where [ ˜ β ( j ( λ ))] denotes the geodesic whose endpoints are the images under ˜ β ◦ j of the endpoints of λ . Proof of Theorem A: conclusion.
We have a geometrically finite Kleinian group Γ with-out parabolics, which does not split as a free product, together with a sequence of strictlytype preserving isomorphisms ρ n : Γ → G n which converge strongly to a purely loxodromicKleinian group G ∞ = ρ ∞ (Γ). Our aim is to use the criterion of Corollary 3.10 to show that thecorresponding sequence of CT -maps ˆ i n : Λ Γ → Λ G n converges uniformly to ˆ i ∞ : Λ Γ → Λ G ∞ . Since the representations ρ n converge strongly to G ∞ , by Proposition 2.2 we may as wellassume that we have a compact core K of N = H / Γ together with embeddings φ n : K (cid:44) → M n = H /G n , n ∈ N ∪ ∞ , such that K n = φ n ( K ) is a Scott core of M n and K = φ ∞ ( K ) is aScott core of M ∞ . We may also choose the lift O of the basepoint o ∈ K and lifts (cid:101) φ n of φ n sothat for all n , ˜ φ n ( O ) lies in a uniformly bounded neighborhood of O ∈ H .We need to show that there exists a function f : N → N such that whenever λ is a d Γ -geodesic segment lying outside B Γ (1; f ( L )) in G Γ, the H -geodesic [ j n ( λ )] lies outside B H ( O ; L )for all n ∈ N , where as usual j n : G Γ → H , j n ( γ ) = ρ n ( γ ) · O .As in 4.2.3 above, let E i , i = 1 , . . . , r be the ends of M = M ∞ . Given L ∈ N , choosecompact submanifolds E i ⊂ E i homeomorphic to S i × [0 ,
1] such that d M (( E i \ E i ) , K ) ≥ L ,where d M denotes the metric induced by shortest paths in M . Identifying M with the model M ,pick q ∈ N so that E i is contained in the union of the first q blocks of E i for each i = 1 , . . . , r .Then with the notation of Proposition 4.9, we have d M ( M \ B ( q ) , K ) ≥ L .By strong convergence, there exists n = n ( L ) ∈ N such that for all n ≥ n , there existsa 2-bi-Lipschitz embedding ψ n : B ( q + D ) → M n , and such that ψ n ( B ( q + D )) ⊃ K n = ψ n ( K )cuts off the ends of M n , see Lemma 2.1. Let o ∈ K be a base-point which lifts to O ∈ H . Upto adjusting by a bounded distance, we may assume that ψ n lifts to a map ˜ ψ n with ˜ ψ n ( O ) = O ,and that ˜ ψ n ( O ) projects to the base-point o n ∈ M n and ψ n ( o ) = o n .Now apply Corollary 4.10 with M = M ∞ and the integer q , and with V = M n and β = ψ n ,to see there exists a function f : N → N , independent of n , with the following property. Let λ be a d Γ -geodesic segment in G Γ lying outside B Γ (1 , f (2 L )). Since ( ψ n ) ∗ ◦ ρ ∞ = ρ n we have (cid:101) ψ n ◦ j ∞ = j n . Let [ j n ( λ )] be the H -geodesic segment with the same endpoints as j n ( λ ). Thenby Corollary 4.10, [ j n ( λ )] ∩ (cid:101) ψ n ( (cid:103) B ( q )) lies outside B H ( O, L ) ⊂ H , where (cid:103) B ( q ) ⊂ (cid:102) M ∞ is thelift to (cid:102) M ∞ = H of B ( q ).Let π : H → M n be the covering projection so that in particular π ( O ) = o n . Since ψ n ( B ( q ))cuts off the ends of M n , then since ψ n is 2-bi-Lipschitz, any point in π ([ j n ( λ )]) which lies outside ψ n ( B ( q )) must be at least distance L to o n . It follows that whenever n ≥ n , [ j n ( λ )] lies outside B ( (cid:101) ψ n ( O ) , L ) ⊂ (cid:102) M n = H whenever λ lies outside B Γ (1 , f ( L )) in G Γ.It remains only to deal with n < n . By Corollary 3.10, for each n ∈ { , . . . , n } , thereexists N n = N n ( L ) ∈ N , such that if λ is a d G -geodesic outside B Γ (1 , N n ( L )) in G Γ, then [ j n ( λ )]lies outside B ( O, L ) ⊂ H . Choosing f ( L ) = max { f ( L ) , N ( L ) , . . . , N n ( L ) } we have verifiedthe criterion of Corollary 3.10. This completes the proof of Theorem A. IMITS OF LIMIT SETS II 23 Algebraic limits and non-convergence of limit points
In this section we prove Theorem C, of which Theorem B is an immediate consequence.The sequence of groups G n in Theorem C is that described by Brock in [9], in which theconvergence is algebraic but not strong. We begin with a brief description of the examples andBrock’s bi-Lipschitz models for the manifolds involved.5.1. Brock’s Examples.
The groups G n in Theorem C are a sequence of quasi-Fuchsiansurface groups converging algebraically but not strongly to a partially degenerate geometricallyinfinite surface group G ∞ with an accidental parabolic. The examples are also discussed brieflyin [30].The sequence G n is obtained as follows. Fix a closed hyperbolic surface X = H / Γ. Let σ be a simple closed geodesic which separates X into two subsurfaces R and L . Let α de-note an automorphism of X such that α | L is the identity and α | R = χ is a pseudo-Anosovdiffeomorphism of R preserving the boundary σ . (For later reference, it is important to ensurethat there is no Dehn twisting around σ when χ is considered as the restriction of α to π ( R ),see 5.3.1 below.) Let G n be the quasi-Fuchsian group given by the simultaneous uniformiza-tion of ( α n ( X ) , X ), so that G n = ρ n (Γ) for suitably normalized ρ n : Γ → SL (2 , C ) and G isFuchsian. This means that the regular set Ω n of G n has two components Ω ± n where the ‘lower’component Ω − n / Γ is conformally equivalent to X and the ‘upper’ component Ω + n / Γ is equivalentto α n ( X ). The algebraic limit G ∞ of the groups G n is a partially degenerate geometrically infi-nite surface group, while with suitable choice of basepoint, the geometric limit of the manifolds M n = H /G n is homeomorphic to X × R \ R × { } . These assertions will be explained inmore detail below. Since to fully understand our example, it is important to be clear about thenotational conventions, we begin by setting these out. As far as possible, we follow Brock [9].5.1.1. Teichm¨uller space and the mapping class group.
Let S be an oriented hyperbolizablesurface. The Teichm¨uller space Teich( S ) of S parametrizes finite area hyperbolic structureson S up to isotopy. Thus a point X ∈ Teich( S ) is a hyperbolic surface X equipped witha homeomorphism f : S → X which marks X . The mapping class group Mod( S ) acts onTeich( S ): if α ∈ Mod( S ) then φ ( S, f ) = (
S, f ◦ α − ). The map on surfaces induces an actionon the space of representations ρ : π ( S, x ) → P SL ( C ) by α ( ρ ) = ρ ◦ α − .If ζ : [0 , → S is a path in S , we denote by α ( ζ ) the path α ◦ ζ : [0 , → S . Thus defininga hyperbolic surface X in terms of the lengths (cid:96) X ( γ ) of the simple closed curves γ on X andidentifying X with S by taking f = id, we can write(1) (cid:96) α ( X ) ( γ ) = (cid:96) X ( α − ( γ )) . Hence the map α : X → α ( X ) is an isometry. The action on curves extends to an action onthe space of measured laminations ML( S ) on S : for µ ∈ ML( S ) the lamination α ( µ ) is definedby (geometric) intersection numbers:(2) i ( γ, α ( µ )) = i ( α − ( γ ) , µ )for all γ ∈ π ( S ). Thus if α ∈ Mod( S ) , µ ∈ ML( S ) and X ∈ Teich( S ) we have (cid:96) α ( X ) ( µ ) = (cid:96) X ( α − ( ν )) where (cid:96) X ( µ ) denotes the length of the lamination µ in the surface X .5.1.2. Quasi-Fuchsian groups.
A quasi-Fuchsian group G is the image of a discrete faithful rep-resentation ρ : π ( S, x ) → P SL ( C ), whose domain of discontinuity has two simply connectedcomponents Ω ± . By convention we take Ω + to be the component whose orientation is the sameas that of S . By Bers’ simultaneous uniformisation theorem, a pair of points X, Y ∈ Teich( S )parametrize quasi-Fuchsian groups: if G = G ( X, Y ) = ρ ( π ( S, x )) then Q ( X, Y ) denotes themanifold for which Ω + /G is conformally equivalent to X and Ω − /G is anti-conformally equiv-alent to Y . Strictly this only defines G ( X, Y ) up to conjugation.To mark Q ( X, Y ) and fix G ( X, Y ) we proceed as follows. Fix a base point s ∈ S . Let Y be the point ( S, f ) ∈ Teich( S ) and let y = f ( s ). Fix a lift ˜ f : H → Ω − which descends to f ,where we identify H with the universal cover of S and where O = O ∈ H is a fixed basepointwhich descends to s , and set ˜ y = ˜ f ( O ). Now the convex hull C of Q ( X, Y ) is bounded by twopleated surfaces ∂ C ± which (when positively oriented, that is, so that ∂ C + is oriented pointingout of C and ∂ C − is oriented pointing into C ) are respectively uniformly bounded distanceto X, Y in Teich( S ). There is a natural retraction map r from Ω − to the lift (cid:103) ∂ C − to H of ∂ C − [16]; we arrange that r (˜ y ) = O = O ∈ H . This gives a map ˜ f : ( H , O ) → ( (cid:103) ∂ C − , O )which descends to a map S → Q ( X, Y ) which sends S to a pleated surface in Q ( X, Y ) atuniformly bounded Teichm¨uller distance to Y . We use this marking to induce the representation ρ : π ( S, s ) → G ( X, Y ). We fix the basepoint o ∈ Q = H /G to be the projection of the point O ∈ H .5.1.3. Iteration of pseudo-Anosovs.
For details on measured laminations and pseudo-Anosovmaps, see [19, 41]. Here is a summary of what we need. Let χ ∈ Mod( S ) be pseudo-Anosov.Then χ has two fixed points in the space P M L ( S ) of projective measured laminations on S : the stable lamination λ s and an unstable lamination λ u . (If necessary, we distinguish theunderlying lamination | λ | from its transverse measure λ .) This means there exists c > χ ( λ s ) = 1 c λ s and χ ( λ u ) = cλ u . From (2) this gives i ( χ − ( γ ) , λ u ) = i ( γ, χ ( λ u )) = ci ( γ, λ u ).Thus lim n →∞ i ( χ n ( γ ) , λ u ) → λ u is uniquely ergodic),(3) [ χ n ( γ )] → [ λ u ] in PML(S) IMITS OF LIMIT SETS II 25 where [ µ ] denotes the projective equivalence class of µ ∈ M L ( S ) in P M L ( S ).Let X be a fixed surface in Teich( S ). The boundary leaves of laminations | λ u | , | λ s | de-compose X into a collection of rectangles in each of which we have a metric (cid:112) ( λ u ) + ( λ s ) (or more simply the equivalent metric λ u + λ s ). Putting these together gives a metric quasi-isometric to the hyperbolic metric on X . Then (cid:96) X ( γ ) ∼ i ( γ, λ u ) + i ( γ, λ s ) and for any arc T we have (cid:96) X ( T ) ∼ i ( T, λ u ) + i ( T, λ s ), where ∼ denotes equality up to multiplicative boundedconstants. In particular, if T is an arc along an unstable leaf then (cid:96) X ( T ) ∼ i ( T, λ s ) so that (cid:96) χ ( X ) ( T ) ∼ i ( χ − ( T ) , λ s ) = c − i ( T, λ s ) ∼ c − (cid:96) X ( T ). In other words, φ contracts along unstableleaves. Also observe that since(4) (cid:96) χ n ( X ) ( γ ) = (cid:96) X ( χ − n ( γ )) ∼ i ( γ, χ n λ s ) + i ( γ, χ n λ u ) = c n i ( γ, λ u ) + c − n i ( γ, λ s ) , it follows by taking ratios of lengths that χ n ( X ) → [ λ u ] in P M L ( S ), viewed as the Thurstoncompactification of Teich( S ).5.1.4. The algebraic limit for iteration of pseudo-Anosov maps.
Given a surface S and α ∈ Mod( S ), the mapping torus of ( S, α ) is the manifold T α = S × [0 , / ∼ where ∼ is theequivalence relation ( x, ∼ ( α ( x ) , N α = S × R be the cyclic cover of T α , correspondingto the subgroup π ( S ). The manifold N α is naturally oriented by the orientation of S .If χ ∈ Mod( S ) is pseudo-Anosov, then Thurston showed that T χ and hence N χ has a hy-perbolic structure [41, 26]. Pick (Σ , f ) ∈ Teich( S ) and consider the manifold M χ correspondingto the algebraic limit of the quasi-Fuchsian groups G ( χ n (Σ) , Σ), where Q ( χ n (Σ) , Σ) is markedas described above. McMullen [26] Theorem 3.11 shows that the limit manifold M χ has onepositive degenerate end E which is asymptotically isometric to the positive end of N χ . (Notethat the positive end of N χ is, up to complex conjugation, the negative end of N χ − , see [26]Proposition 3.10.)The end E = E (Σ , χ ) consists of successive sheets which are mapped one to the next by χ .More precisely, we have a sequence of pleated surfaces h j : S → E exiting E such that the j th level surface is marked by the map f ◦ χ − n : S → M χ . In particular h : S → Σ × { } is the map h ( x ) = ( f − ( x ) ,
0) for x ∈ Σ; loosely, h identifies S with Σ = Σ × { } . Up to quasi-isometry, E is modelled by Σ × [0 , ∞ ) with the image of the j th level pleated surface Σ j identified withΣ × { j } . The hyperbolic structure of Σ j is the point χ n (Σ) ∈ Teich( S ). Now put a metric onΣ × [0 ,
1] which smoothly interpolates between Σ and Σ and then transport this metric toΣ × [ i, i + 1] using the isometry χ i . This gives a uniformly bi-Lipschitz homeomorphism from E to the model manifold Σ × [0 , ∞ ). The model is marked by the map h ◦ f which sends abase-point s ∈ S to a base-point o = ( f ( s ) , ∈ Σ The convex cores of the approximating manifolds Q ( χ n (Σ) , Σ) are equally modelled byΣ × [0 , n ] with the restriction of the above metric. The above marking is, up to a uniformlybounded discrepancy, the same as the one described in 5.1.2, and hence determines the limitrepresentation ρ : π ( S, s ) → π ( M χ , o ). Lemma 5.1 ([9] Lemma 4.4) . The ending lamination of the end E of M χ is the unstablelamination λ u of χ .Proof. This is proved in [9]. Here is a variant which will serve as a check we have the correctconventions. Think of the model E = Σ × [0 , ∞ ) as quasi-isometrically embedded in M χ . Let s be the base-point in S . As above, with (Σ , f ) ∈ Teich( S ) we have base-point o = ( f ( s ) , ∈ Σ ⊂ M χ . A loop γ ∈ π ( S, s ) defines a path ( f ◦ γ, ⊂ Σ and hence a homotopy class[ ρ ( γ )] ∈ π ( M χ , o ). Now consider the path χ n ( γ ) = χ n ◦ γ ∈ π ( S ). To find the approximatelength of the geodesic in the class [ ρ ( χ n ( γ ))] in M χ , let τ n be the path t (cid:55)→ ( f ( s ) , t ) , t ∈ [0 , n ] ⊂ E . Note that ( f ◦ χ n ( γ ) ,
0) is homotopic in M χ to the loop τ n ( f ◦ χ n ( γ ) , n ) τ − n and hence freelyhomotopic to the path ( χ n ( γ ) , n ) ⊂ Σ n .Now by (1), (cid:96) Σ n ( χ n ( γ )) = (cid:96) χ n (Σ) ( χ n ( γ )) = (cid:96) Σ ( γ ). Thus the sequence of curves ( χ n ( γ ) , n ) onthe pleated surfaces Σ n exit the positive end of M χ and have uniformly bounded length. Thismeans they converge to the ending lamination of M χ . On the other hand, by (3), [ χ n ( γ )] → [ λ u ]in P M L (Σ). Hence the ending lamination of M χ is λ u . (cid:3) The algebraic limit for iteration of partially pseudo-Anosov maps.
Now we turn to thecase under consideration, in which α ∈ Mod S is partially pseudo-Anosov as described inSection 5.1 above. Thus S is now a closed surface separated into two components R and L by a simple closed curve σ and α ∈ Mod( S ) is such that α | L is the identity and α | R = χ is apseudo-Anosov diffeomorphism of R preserving σ .Given X ∈ Teich( S ) we set G n = G ( α n ( X ) , X ) and M n = Q ( α n ( X ) , X ), so that M n is themanifold such that Ω + /G n is conformally equivalent to α n ( X ) while Ω − /G n is anti-conformallyequivalent to X . In particular, G is a Fuchsian group uniformizing X and our sequence ofrepresentations are the maps ρ n : G → G n .Brock showed, [9] Theorem 5.4, that the representations ρ n converge algebraically to arepresentation ρ ∞ : G → G ∞ where G ∞ is a geometrically infinite surface group with corre-sponding manifold M ∞ . The regular set Ω ∞ of G ∞ has one G ∞ -invariant simply connectedcomponent Ω −∞ such that Ω −∞ /G ∞ is conformally equivalent to X . The positive end (corre-sponding to Ω + ) has degenerated: if g σ ∈ π ( S ) corresponds to the separating curve σ , then ρ ∞ ( g σ ) is an accidental parabolic and Ω + ∞ collapses to a countable collection of simply con-nected components Ω + ,i whose stabilisers are each conjugate to ρ ∞ ( π ( L )), so that Ω + ,i /G ∞ is IMITS OF LIMIT SETS II 27 a Riemann surface topologically equivalent to Int L for each i , with ρ ∞ ( g σ ) representing a loopencircling a puncture. Correspondingly, the upper end of M ∞ is partially degenerate; the partcorresponding to L is geometrically finite while the part corresponding to R is degenerate withending lamination the unstable lamination of χ .Let H be a horocyclic neighborhood of the cusp corresponding to ρ ∞ ( g σ ) in M ∞ . Theassertion of [9] Theorem 5.4 is that the end of M ∞ \ H cut off by the surface R is asymptoticallyisomorphic to the end E (Σ , χ ) described in the previous section, where Σ is a hyperbolic surfacewith the same topology as Int R but equipped with a complete hyperbolic structure so that theboundary σ = ∂R is replaced by a cusp on Σ and χ = α | R .5.1.6. Models of the approximating manifolds.
Minsky [29] § M n = Q ( α n ( X ) , X ) in terms of a uniformly bi-Lipschitz model for its convexcore C n . Fix X ∈ Teich( S ) such that (cid:96) X ( σ ) < (cid:15) for some (cid:15) less than the Margulis constant.As above σ separates X into surfaces R, L ; when needed we distinguish between the topologicalsurfaces
R, L and the hyperbolic structures X R , X L induced from X .Also pick a complete hyperbolic surface Σ with the same topology as Int R but so thatthe boundary σ = ∂R is replaced by a cusp on Σ. Let Σ c denote Σ with a small (open)neighborhood of the cusp removed so that the boundary curve, which we denote σ c , has length (cid:15) . First we make a model B n for the part of C n corresponding to R . Let N = N χ be thehyperbolic 3-manifold with fiber Σ and monodromy χ . Let N n denote the cyclic n -fold coverof N , i.e. the manifold whose fundamental group is the kernel of the homomorphism π ( N ) → π ( S ) = Z → Z n . Let N cn denote N n with a small (open) neighborhood of the cusp removed sothat the boundary curve of each fiber has length (cid:15) . Let ˆ N cn be the manifold obtained by cutting N cn open along a lift of some fiber and completing metrically to a manifold with boundary. Thenjust as described in Section 5.1.4, there is a uniformly bi-Lipschitz homeomorphism from ˆ N cn toa model manifold B n = Σ c × [0 , n ], in which the block Σ c × [ i, i + 1] is isometric to Σ c × [0 ,
1] bya map homotopic to χ − i . In the model metric on B n , the boundary loop σ c × { t } has length (cid:15) for each t ∈ [0 ,
1] and the boundary cylinder σ c × [0 , n ] has the obvious product of the Euclideanmetrics on σ c and the interval [0 , n ]. (The discussion in [7] §
8, especially Proposition 8.6 forthe discussion of ∂ Σ, explains the model for N cn in a neighborhood of a puncture.)The model of the part of C n corresponding to L is essentially the product metric on C = X L × [0 , X L slightly to ensure the boundary circles ∂L × { t } allhave length (cid:15) and take the standard Euclidean metric of length one on the second factor. Now we can make model K n for the whole of C n . Glue the circle ∂L × { } ⊂ C n to the circle ∂R × { } = σ c × { } ⊂ B n , and likewise glue the circle ∂L × { } to the circle σ c × { n } . Let K cn denote the resulting space. Let η denote a circle of length ( n + 1) obtained by moving in thedirection of the second factor in K cn . Finally, let K n be the manifold (with boundary) obtainedby (hyperbolic) Dehn filling K cn with a Margulis tube T n with meridian η and longitude σ c ,smoothing out at the boundary if needed. Figure 2 shows a ‘cross-section’ of K n . Note thatthe manifolds K n have uniformly bounded geometry away from T n .The lower boundary of K n , denoted ∂K − n , is obtained by gluing ∂L × { } to ∂R × { } . Thuswe have an obvious map φ n : S → ∂K − n which we use to mark K n . If s ∈ σ is the base-pointof S , we denote the image φ n ( s ) = ( s , ∈ ∂L × { } by o n . Lifting everything to universalcovers, identifying (cid:101) S with H , and thinking of (cid:101) K n ⊂ H , we can arrange that s lifts to O ∈ H and o n lifts to O n = O ∈ H .The upper boundary of K n , denoted ∂K + n , is obtained by gluing ∂L × { } to ∂R × { n } .This gives a second obvious embedding φ + n : S → ∂K + n and we write o + n = φ + n ( s ), with lift O + n ∈ H . Occasionally we write ø − n for o n , O − n for O n = O and φ − n for φ n for clarity.To see that the model manifolds K n are bi-Lipschitz equivalent to the convex cores C n ,note that the marked surfaces ∂K ± n of K n are conformally a uniformly bounded Teichm¨ullerdistance from the surfaces α n ( X ) , X respectively, precisely as in the case of the manifolds M n = Q ( α n ( X ) , X ). Thus the standard techniques used in the proof of the ending laminationtheorem show that there are bi-Lipschitz homeomorphisms between C n and K n , with constantsuniform in n . These are the models we will use.5.1.7. The limit manifolds.
In the algebraic limit, the tube T n becomes a rank one cusp, seealso [30]. The lower boundary ∂K − n of K n stays fixed but the upper boundary ∂K + n developsinto a partially degenerate end in which L becomes a surface with a puncture. The part of thesurface corresponding to R becomes the degenerate end E described in 5.1.6.In the geometric limit, the distance from o − n ∈ ∂K − n to o + n ∈ ∂K − n stays bounded, becausewe can always travel through the bounded half C = L × [0 , c × { n/ } of B n we have to travel ever further, either going directly ‘up’ through B n or through L × [0 , ∂L × { } , and then ‘down’ from Σ c × { n } through B n toΣ c × { n/ } . Thus in the geometric limit, K n converges to the manifold S × R \ R × { } withtwo geometrically infinite ends, each asymptotically quasi-isometric to E . This is discussed indetail in [9], but is not important for us here.5.2. Absence of Uniform Convergence.
Since the sequence G n does not converge strongly,by Proposition 3.7, it must fail to satisfy UEP. In fact it is easy to exhibit a sequence of points IMITS OF LIMIT SETS II 29
Figure 2.
A schematic picture of K built up of B , C and T . The points a, b are the base-points φ − n ( s ) = o − n and φ + n ( s ) = o + n respectively. g k ∈ G such that | g k | → ∞ while d ( O, ρ n ( g k ) · O ) ≤ c for all n, k and some fixed c >
0. (Tosee that this is equivalent to violating UEP, see [38] Lemma 5.2.)The meridian curve η round the boundary of the Margulis tube T n is split into two homo-topic paths τ n , υ n in K n by the points o ± n ∈ ∂K ± n . The path τ n : t (cid:55)→ ( s , t ) , t ∈ [0 , n ] joins o ± n going the ‘long’ way round ∂T n in B n , while the path υ n : t (cid:55)→ ( s , t ) , t ∈ [0 ,
1] goes the ‘short’way round in C n .Let s (cid:55)→ γ ( s ) be a based loop homotopic to a fixed generator of π ( S, s ) and lying entirelyon R . As in the proof of Lemma 5.1, the path γ n : [0 , → K n , γ n ( s ) = ( χ n γ ( s ) , n ) ∈ Σ c ×{ n } ⊂ ∂K + n has the same length (cid:96) say as the path s (cid:55)→ γ ( s ) = ( γ ( s ) , ∈ Σ c × { } ⊂ ∂K − n .Now the loops τ n γ n τ − n and υ n γ n υ − n are homotopic in K n , moreover from the above obser-vation, υ n γ n υ − n has length 2 + (cid:96) in K n . On the other hand in K n , τ n γ n τ − n is homotopic to theloop χ n ( γ ) ⊂ Σ c × { } ⊂ ∂K − n , where by χ n ( γ ) we mean the path s (cid:55)→ ( χ n ( s ) , ∈ Σ c × { } .By (4), the geodesic length of χ n ( γ ) on Σ c × { } increases exponentially with n ; hence bythe usual comparison of word length and geodesic length on Σ c × { } , if g n ∈ G repre-sents the loop χ n ( γ ) ∈ π ( S, s ) then | g n | → ∞ in G . Since ρ n is induced by the marking φ n : s (cid:55)→ ( s, ∈ K − n , the loop χ n ( γ ) is in the homotopy class of ρ n ( g n ) ∈ π ( K n ; o − n ).Let O be the lift to H of o − n ∈ K n as above. Lifting the paths υ n γ n υ − n , we have founda sequence g n ∈ G for which d G (1 , g n ) → ∞ but for which d H ( O, ρ n ( g n ) O ) is uniformlybounded. As noted above, this violates UEP. Pointwise non-convergence.
Let Γ = G be the Fuchsian group for which X = H / Γ.Recall that σ corresponds to a loxodromic g σ ∈ Γ whose image under ρ ∞ is parabolic. Let P ⊂ Λ Γ denote the endpoints of axes which project to σ , equivalently, the set of images underΓ of the fixed points of g σ . The counter examples we seek for Theorem C occur in the case of ξ ∈ Λ Γ for which ξ / ∈ P but ˆ i ∞ ( ξ ) = ˆ i ∞ ( p ) for some p ∈ P . Precisely which points these are isgiven by the following theorem of Bowditch. Theorem 5.2 ([7] Theorem 0.2) . Let H / Γ be a punctured hyperbolic surface. Let M = H /G be a simply degenerate hyperbolic manifold corresponding to a faithful type preserving represen-tation ρ : Γ → G , and suppose that there is a lower bound to the length of all loxodromics in M .Suppose that M has ending lamination λ and let ˆ i : Λ Γ → Λ G be the corresponding CT -map.Then ˆ i ( ξ ) = ˆ i ( η ) , ξ, η ∈ Λ Γ if and only if ξ and η are either either ideal end-points of the sameleaf of λ , or ideal boundary points of a complementary ideal polygon of λ . This result was originally proved by Minsky [28] in the (bounded geometry) closed surfacecase. The condition on loxodromics means of course that the injectivity radius of M is boundedbelow outside a horoball neighborhood of the punctures of S . This result has been extendedto unbounded geometry and more general manifolds in [39], [33], [15].5.3.1. The points of non-convergence.
The points of non-convergence of the maps ˆ i n will be theendpoints of lifts of unstable leaves which bound the crown domain of the unstable lamination | λ u | of χ in Int R . First, as mentioned above, we need to be careful about the precise meaningof saying that α | R is pseudo-Anosov, so as to ensure that there is no Dehn twisting around σ when we consider χ as the restriction of α . We suppose given the hyperbolic surface Σas above and a pseudo-Anosov map χ : Σ c → Σ c which pointwise fixes σ c and which is theidentity in a horoball neighborhood of the cusp Σ \ Σ c . Then χ induces an automorphism χ ∗ of π (Σ c , s ), where we pick s ∈ ∂ Σ c . Now identify R with Σ c and ∂R = σ with ∂ Σ c . Withthis identification, we insist that ( α ∗ ) | π ( R,s ) = χ ∗ .Continuing with the identification of R with Σ c , note that the crown domain F of theunstable lamination λ u of χ is an annulus with one boundary component σ and the otherconsisting of finitely many alternating segments of stable and unstable leaves. By taking asuitable power of χ if necessary, we can assume that these leaves map to themselves under χ .Next, pick a lift ˜ F of F and a corresponding lift ˜ χ of χ which maps ˜ F to itself and which isthe identity on a particular lift ˜ σ of σ . Let µ u be the lift of one of the unstable leaves boundingthe lift ˜ F and let ξ u be one of its endpoints in Λ Γ . From our assumption that ( α ∗ ) | π ( R,s ) = χ ∗ ,it follows that χ ( ξ u ) = ξ u . (Without this assumption, we might have χ ( ξ u ) = ρ ( g σ ) k ( ξ u ) forsome k ∈ Z .) IMITS OF LIMIT SETS II 31
The non-convergence part of Theorem C is proved by
Proposition 5.3.
Let ξ u be an endpoint of a boundary leaf of ˜ F , and let p ∈ P be an endpointof the lift of σ also bounding ˜ F . Then ˆ i ∞ ( ξ u ) = ˆ i ∞ ( p ) , while no subsequence of the sequence ˆ i n ( ξ u ) limits on ˆ i ∞ ( p ) .Proof. The statement that ˆ i ∞ ( ξ u ) = ˆ i ∞ ( p ) follows from Theorem 5.2 since λ u is the endinglamination of M ∞ . The statement that no subsequence of the sequence ˆ i n ( ξ u ) limits on ˆ i ∞ ( p )is Corollary 5.6 which we prove below. (cid:3) To prove Corollary 5.6, we will construct, for each n , a quasi-geodesic in the lift (cid:101) K n of K n which passes through the basepoint O ∈ (cid:102) K n , and with endpoints ˆ i n ( ξ u ) and ˆ i n ( p ). Thesequasi-geodesics will be uniform in n and the result will follow.We want to consider ξ u ∈ Λ Γ as a point in the limit set Λ n Γ of the surface α n ( X ). To dothis, denote by H n the universal cover of the surface α n ( X ), with basepoint ˜ s = ˜ χ n (˜ s ) ∈ ˜ σ .The map ˜ α n : H → H n extends to a homeomorphism h n : Λ Γ → Λ n Γ . Since ˜ α n ( µ u ) = µ u , itfollows that h n ( ξ u ) = ξ u .In H n , let P n be the foot of the perpendicular from O = ˜ s to µ u and consider the path β n which follows the perpendicular from O to P n and then follows µ u from P n to its endpoint ξ u .The segment from O to P n has length bounded independent of n since outside the thin part of X , the diameter of the plaque ˜ F is bounded. Hence β n is quasi-geodesic in H n .The marking of K n is given by the embedding φ n : ( S, s ) → ( ∂K − n , o − n ) which lifts to (cid:102) φ n : ( H , O ) → ( (cid:101) K n , O ). This extends to the map ˆ i n : Λ Γ → Λ n , where Λ n is the limit set of G n . On the other hand, the upper boundary ∂K + n of K n is marked by the map φ + n = φ n ◦ χ − n whose lift (cid:102) φ + n : ( H n , O ) → ( (cid:93) ∂K + n , O + n ) extends to a map q n : Λ n Γ → Λ n . Clearly, q n ◦ h n = ˆ i n , soin particular, q n ( ξ u ) = ˆ i n ( ξ u ). Lemma 5.4.
The path (cid:102) φ + n ( β n ) from O + n to ˆ i n ( ξ u ) is quasi-geodesic in (cid:101) K n , with constants uni-form in n .Proof. The segment of (cid:102) φ + n ( β n ) from O + n to (cid:102) φ + n ( P ) has uniformly bounded length, so it is sufficientto show that (cid:102) φ + n ( µ n ) is uniformly quasi-geodesic in (cid:101) K n .Suppose first that we were dealing with the case of a pseudo-Anosov map χ on a puncturedhyperbolic surface Y , so that the stable and unstable laminations λ s , λ u of χ fill up Y . Let dx denote the transverse measure to λ s and dy denote the transverse measure to λ u , so that dx measures length along unstable leaves and dy measures length along stable leaves. Asin 5.1.3, this defines a singular metric on Y which for brevity we write as ds = dx + dy . Since χ expands along stable leaves, that is in the y -direction, the metric on χ n ( Y ) is given by ds = c − n dx + c n dy . The same formula defines a singular metric on the universal cover H .Restricting the model end E = Y × [0 , ∞ ) in 5.1.3 to E n = Y × [0 , n ] provides a modelfor the convex core of Q ( χ n ( Y ) , Y ). We have obvious maps which embed Y and χ n ( Y ) in E n as pleated surfaces Y = Y × { } , Y n = Y × { n } respectively. Passing to universal covers, asin [28], see also [14], the metric in (cid:101) E n is modelled by ds = dt + c − t dx + c t dy , where t is the ‘vertical’ coordinate in the second factor. Thus the map which projects (cid:101) E n ‘vertically’upwards to (cid:101) Y n = (cid:101) Y × { n } is a contraction when restricted to µ u × [0 , n ], where as above µ u is a boundary leaf of λ u . Hence projecting from (cid:101) E n to ( µ u , n ) by first projecting ‘horizontally’in the surface (cid:101) Y m to ( µ u , m ) and then ‘vertically’ to ( µ u , n ) is a contraction, from which theresult (that a leaf of the unstable lamination on the top surface (cid:101) Y n is quasi-geodesic) followsby standard methods, see for example Bowditch [8] Lemma 4.2.In the present case the model is somewhat more complicated because the automorphism α of the underlying surface S is partially pseudo-Anosov and K n limits on a partially degenerateend of M ∞ . However we can apply the above argument working in the space in which weelectrocute the left hand half C n = L × [0 , n ], together with the Margulis tube T n around σ n .An equivalent proof can be constructed by modelling (cid:101) K n as a tree of hyperbolic metricspaces as in [31]. (cid:3) Next, we modify the path (cid:102) φ + n ( β n ) of the previous lemma to a quasi-geodesic path from O = O − n with the same endpoint ˆ i n ( ξ u ) ∈ Λ n , by prefixing it with the path υ n from O − n to O + n which goes the ‘short’ way round ∂T n in C n as in section 5.2. Since υ n has uniformly boundedlength 1, the resulting path δ n is a (cid:101) K n - quasi-geodesic from O to ˆ i n ( ξ u ). It follows that thegeodesic ray from O to ˆ i n ( ξ u ) either lies completely outside T n , or enters T n only to exit at apoint O (cid:48) a uniformly bounded distance from O . Lemma 5.5.
Let p ∈ P be as in the statement of Proposition 5.3. Let γ n be the hyperbolic rayfrom O to ˆ i n ( p ) , and let δ n be as above. Then the angle at O between γ n and δ n is uniformlybounded away from .Proof. First consider first the limiting case in which g σ is parabolic. After normalizing andworking in the upper half space model H , we may assume we are in the following situation.Let O ∈ H be a fixed base point at Euclidean height 1 above the base plane. Suppose that A ∈ Isom H is a parabolic fixing ∞ . Suppose that H is the height 1 horoball at ∞ , so that O ∈ ∂H . Suppose that δ is a geodesic ray from O which either lies completely outside H , orwhich enters H and leaves it again at a point O (cid:48) at distance at most k from O . Let γ be the IMITS OF LIMIT SETS II 33
Figure 3.
Geodesic realizations. The ray from a = O − n to x (the image of ageodesic ray in (cid:101) S ) lies on the lower boundary ∂ (cid:101) K − n while its geodesic realizationtravels the short way round the (lifted) Margulis tube (cid:101) T to b = O + n and thenceruns along the upper boundary ∂ (cid:101) K + n . The endpoints of the two rays coincide in ∂ H .ray from O to ∞ . Then the angle α between δ and γ at O is bounded away from 0; precisely2 | cot α | ≤ k (cid:48) , where k (cid:48) is the Euclidean bound on distance corresponding to the hyperbolicdistance k .Now we extend to the case of loxodromics of short translation length. Working in the upperhalf space model H , let A n ∈ Isom H be a loxodromic fixing ∞ . Suppose that the translationlength (cid:96) ( A n ) → n → ∞ . Let T n be a constant distance cone around Ax A n , chosen so thatthe translation length of A n restricted to ∂T n is a fixed length (cid:15) . Let O ∈ H be a fixed basepoint normalized to be at height 1 and assume that O ∈ ∂T n . Let a n be the other end pointof Ax A n . Let γ n be the geodesic ray from O to a n and let δ n be another geodesic ray from O which either lies completely outside T n , or which enters T n and leaves it again at a point O (cid:48) athyperbolic distance at most k from O , where k is bounded independent of n . We want to showthat the angle between δ n and γ n at O is bounded away from 0.Let θ n be the angle between the sides of the cone (cid:101) T n and the horizontal. Since (cid:96) ( A n ) → A n restricted to ∂ (cid:101) T n is fixed, θ n → n → ∞ . Now the angle between γ n and the vertical at O is 2 θ n . On the other hand, as is easy to compute, the anglebetween ∂ (cid:101) T n and δ n at O is uniformly bounded away from π/
2. Since ∂ (cid:101) T n is nearly horizontal,this proves the result. (cid:3) Corollary 5.6.
No subsequence of the sequence ˆ i n ( ξ u ) limits on the point ˆ i ∞ ( p ) .Proof. By Lemma 5.5, the visual angle subtended by ˆ i n ( p ) and ˆ i n ( ξ u ) at O is uniformly boundedbelow away from 0. Since p is the fixed point of an element of Γ, by algebraic convergence wehave lim n →∞ ˆ i n ( p ) = ˆ i ∞ ( p ) and the result follows. (cid:3) Remark 5.7.
In hindsight, Proposition 5.3 is perhaps not too unexpected as the paths ˜ φ + n ( β n )live on the top sheet of the approximating manifolds (cid:101) K n and these converge to a ray whoselimit does not lie in the limit set of the original surface subgroup π ( S ).5.4. Pointwise Convergence.
We shall now establish that the CT -maps ˆ i n : Λ Γ → G n of theBrock examples converge pointwise for all points ξ ∈ Λ Γ other than those described in Propo-sition 5.3. We do this by applying the conditions EP ( ξ ) (Embedding of Points) and EP P ( ξ )(Embedding of Pairs of Points) for pointwise convergence from [38]. These are essentially thecriteria UEP and UEPP, relaxed so as to allow for dependence on the limit point ξ .5.4.1. Convergence criteria.
In [38] we described EP ( ξ ) and EP P ( ξ ) in relation to a sequenceof elements g i ∈ Γ chosen so that g i · O is a quasi-geodesic in G Γ and so that g i · O → ξ in theEuclidean metric on the ball model B ∪ ∂ B . It is easily seen that is equivalent to replace thiswith a criterion on the geodesic ray [ O, ξ ) from O to ξ in the universal cover H of X = H / Γ,where X ∈ Teich( S ). Definition 5.8.
Let Γ be a Fuchsian group such that X = H / Γ is a closed hyperbolic surfaceand let ρ n : Γ → G n be a sequence of isomorphisms to Kleinian groups G n . Suppose givena sequence of (Γ , G n ) -equivariant embeddings ˜ φ n : ( H , O ) → ( H , O ) which induce basepointpreserving embeddings φ n : X → M n with ( φ n ) ∗ = ρ n . Let ξ ∈ Λ Γ and let [ O, ξ ) be the geodesicray in (cid:101) X = H as above. (1) The pair (( ρ n ) , ξ ) is said to satisfy EP ( ξ ) if there exist functions f ξ : N → N and M ξ : N → N , with f ξ ( N ) → ∞ as N → ∞ , such that for all x ∈ [ O, ξ ) outside B ( O, N ) in H , (cid:101) φ n ( x ) is outside B ( O, f ξ ( N )) in H , for all n ≥ M ξ ( N ) . (2) The pair (( ρ n ) , ξ ) satisfies EP P ( ξ ) if there exists a function f (cid:48) ξ ( N ) : N → N such that f (cid:48) ξ ( N ) → ∞ as N → ∞ , and such that for any subsegment [ x, y ] ⊂ [ O, ξ ) lying outside B ( O ; N ) in H , the H -geodesic [ (cid:101) φ n ( x ) , (cid:101) φ n ( x )] lies outside B ( O ; f (cid:48) ξ ( N )) in H for all n ≥ M ξ ( N ) , where M ξ is as in (1). IMITS OF LIMIT SETS II 35
Note that in these definitions, we do not assume that M ξ ( N ) → ∞ with N , in fact in thebest situation, M ξ ( N ) = 1. We have: Theorem 5.9 ([38] Theorem 7.3) . Suppose that ρ n : Γ → P SL ( C ) is a sequence of discretefaithful representations converging algebraically to ρ ∞ : Γ → P SL ( C ) , and suppose the corre-sponding CT -maps ˆ i n : Λ Γ → Λ G n exist, n = 1 , . . . , ∞ . Let ξ ∈ Λ Γ . Then ˆ i n ( ξ ) → ˆ i ∞ ( ξ ) as n → ∞ if (( ρ n ); ξ ) satisfies EP P ( ξ ) . Verifying pointwise convergence.
As above, we take Γ = G and ρ n : G → G n to be theBrock examples as in Section 5.1.5. Sometimes it will be important to distinguish between thesurface S and the hyperbolic structure X ∈ Teich( S ). Fixing such a structure X , we may takethe dividing curve σ to be geodesic on X . The restrictions X R , X L of X to R, L are hyperbolicsurfaces with geodesic boundary σ . The universal cover (cid:101) S of S is identified with H using thelift of the structure X . Since K n is a quasi-isometric model for the convex core of Q ( χ n ( X ) , X ),we can identify the universal cover (cid:101) K n of K n with a convex subset of H .The representations ρ n correspond to a sequence of embeddings (cid:101) φ n : ( H , O ) → ( H , O )which descend to the maps φ n : ( S, s ) → ( X × { } , ( x , ⊂ K n . The map (cid:101) φ n extends tothe CT -map ˆ i n : Λ Γ → Λ n . Hence if ξ ∈ Λ G , the ray [ O, ξ ) ⊂ H maps under (cid:101) φ to a path (cid:101) φ ([ O, ξ )) ⊂ H joining (cid:101) φ ( O ) = O to ˆ i n ( ξ ) ∈ Λ n . We denote the H -geodesic with these endpointsby [ O, ˆ i n ( ξ )). We will prove convergence ˆ i n ( ξ ) → ˆ i ∞ ( ξ ) by checking that ( ρ n , ξ ) satisfies thecondition EP P ( ξ ).5.4.3. Electric metrics.
For ξ ∈ Λ G , there are two possibilities for the geodesic ray [ O, ξ ) ⊂ (cid:101) S :either it is eventually contained in a fixed lift of R , or not. In the first case, translating by anappropriate element of Γ = π ( S ) we may assume without loss of generality that the entire ray[ O, ξ ) lies in a fixed lift of R .Now consider the model manifold K n and let D n = B n ∪ T n ⊂ K n . Since Margulis tubesare convex and since the tube T n separates B n from C n , it follows that each lift (cid:102) D n of D n isuniformly quasi-convex in (cid:102) K n . Hence ( (cid:102) K n , D n ) satisfies the conditions of Lemma 2.6, where D n is the collection of lifts (cid:102) D n of D n . Let d ne denote the resulting electric metric on (cid:102) K n with thecollection D n electrocuted.Since the curve σ along which we cut X is geodesic, the lifts to H of X R are convex,moreover they are clearly uniformly separated. Let d Se denote the induced electric metric on H with lifts of X R electrocuted. Clearly the ray [ O, ξ ) ⊂ H has infinite length in d Se if andonly if it is not eventually contained in a fixed lift of R . Lemma 5.10.
The map (cid:101) φ n : ( (cid:101) S, d Se ) → ( (cid:102) K n , d ne ) is a quasi-isometry with constants which areuniform in n . Proof.
The map (cid:101) φ n is the lift to (cid:101) S of the map which sends x ∈ L to ( x, ∈ X L × [0 ,
1] and x ∈ R to ( x, ∈ X R × [0 , n ]. The lifts of the complement of X L are electrocuted in (cid:101) S and thelifts of the complement of X L × [0 ,
1] are electrocuted in (cid:102) K n . This result follows since X L × [0 , K n . (cid:3) Corollary 5.11.
The geodesic ray [ O, ˆ i n ( ξ )) ⊂ (cid:101) K n has infinite length in the electric metric d ne if and only if the ray [ O, ξ ) ⊂ (cid:101) S is not eventually contained in a fixed lift of R . In the light of this corollary, the property of the ray [ O, ˆ i n ( ξ )) having infinite d ne -lengthdepends only on ξ . Thus we have two cases to consider depending on whether [0 , ξ ) has finiteor infinite length in the metric d Se . In both cases, to prove convergence ˆ i n ( ξ ) → ˆ i ∞ ( ξ ), we willverify EP P ( ξ ).5.4.4. Case 1: The length of [0 , ξ ) in the electric metric d Se is infinite. Let ξ ∈ Λ Γ . First weprove EP ( ξ ). Since [ O, ξ ) has infinite d Se -length, no tail of [ O, ξ ) is contained in a lift of X R .Hence [ O, ξ ) either crosses X L infinitely often, or has an infinite tail ending in a single lift of X L . It follows that there exists a proper function f ξ : N → N such that if x ∈ [0 , ξ ) is atdistance at least N from O in (cid:101) S , then d Se ( O, x ) ≥ f ξ ( N ).By Lemma 5.10, the map (cid:101) φ n is a uniform quasi-isometry with respect to the respectiveelectric metrics. Hence d ne ( O, (cid:101) φ n ( x )) ≥ c (cid:48) f ξ ( N ) for some constant c (cid:48) >
0. Now any two lifts of C n = X L × [0 ,
1] in (cid:101) K n are separated by a constant c > n . Hence d H ( u, v ) ≥ cd ne ( u, v ) for any points u, v ∈ (cid:101) K n . Absorbing the constants c, c (cid:48) into the function f ξ , we have shown that d H ( O, (cid:101) φ n ( x )) ≥ f ξ ( N ), which is just the statement EP ( ξ ).Now we prove EP P ( ξ ). Consider a segment [ a, b ] ⊂ [ O, ξ ) such that d ( O, y ) ≥ N for all y ∈ [ a, b ]. From the above, d H ( O, (cid:101) φ n ( y )) ≥ f ξ ( N ) for all y ∈ [ a, b ]. In other words, (cid:101) φ n ([ a, b ])lies outside B ( O, f ξ ( N )) in (cid:101) K n .Now replace [ a, b ] by the electro-ambient geodesic with the same endpoints. By Lemma 2.6 ,this is a bounded distance from [ a, b ]. It follows that [ a, b ] is an electric quasi-geodesic in ( (cid:101) S, d Se ).Hence by Lemma 5.10, (cid:101) φ n ([ a, b ]) is an electric quasi-geodesic in ( (cid:101) K n , d ne ) with constants whichare uniform in n .By Lemma 2.6 again, the d ne -electro-ambient geodesic obtained by replacing intersections of (cid:101) φ n ([ a, b ]) with the lifts (cid:101) D n by hyperbolic geodesics in (cid:101) D n with the same endpoints, is a uniformhyperbolic quasi-geodesic in (cid:101) K n with the hyperbolic metric. Hence (cid:101) φ n ([ a, b ]) is a boundeddistance away from the H -geodesic with the same endpoints. This proves EP P ( ξ ). IMITS OF LIMIT SETS II 37
Case 2: The length of [0 , ξ ) in the electric metric d Se is finite. As before, let
P ⊂ Λ G denote the set of endpoints of lifts of the geodesic σ and let P ∞ ⊂ Λ ∞ be the image of thesepoints under ˆ i ∞ . We will prove Proposition 5.12.
If the length of [0 , ξ ) in the electric metric d Se is finite and if ˆ i ∞ ( ξ ) / ∈ P ∞ ,then lim N →∞ ˆ i n ( ξ ) = ˆ i ∞ ( ξ ) . This will complete the proof of Theorem C. This proposition is the only point at which weuse Theorem 4.2.As above, Σ is a surface with the same topology as Int R but equipped with a completehyperbolic structure so that the boundary curve σ is replaced by a puncture on Σ. We shallprove Proposition 5.12 by comparison with the behaviour of CT -maps for the sequence ofquasi-Fuchsian groups F n uniformizing ( χ n (Σ) , Σ) with corresponding manifolds Q ( χ n (Σ) , Σ),so that in particular, F is Fuchsian and H /F = Σ. (Here χ is the same pseudo-Anosov map α | R as above, extended as the identity map in a neighbourhood of the puncture on Σ.) As inSection 5.1.4, up to possibly passing to a subsequence, these groups limit on a simply degenerategroup F ∞ with corresponding manifold M χ whose ending lamination is the unstable laminationof χ . By [26] Theorem 3.12 the convergence is strong. Hence by Theorem 4.2 the CT -mapsˆ k n : Λ F → Λ F n converge uniformly to the CT -map ˆ k ∞ : Λ F → Λ F ∞ . Thus the sequence ofrepresentations ¯ ρ n : π (Σ) → F n satisfies U EP P . As above, we can model the convex coresof the manifolds Q ( χ n (Σ) , Σ) by the restriction E n = Σ × [0 , n ] of the end E of M χ . Thesemodel manifolds E n are marked by the embedding ψ n : Σ → Σ × { } , which lifts to base pointspreserving embeddings (cid:101) ψ n : ( (cid:101) Σ , O ) → (cid:94) (Σ × { } , O ) ⊂ (cid:101) E n .To use the comparison between the representations ¯ ρ n of F = π (Σ) and ρ n of G = π ( S ),we need to make definite the precise relationship between the limit sets Λ F and Λ G . Bydefinition the component Ω − ( G ) of the regular set of G projects to the Riemann surface X .Let (cid:101) R be a fixed component of the lift of R to Ω − ( G ), and let J ⊂ G be its stabiliser, withcorresponding limit set Λ J ⊂ Λ G . Then (cid:101) R /J can be identified with X R so that J = π ( R ). Let V be a bi-Lipschitz homeomorphism X R → Σ c . Clearly V induces an isomorphism V ∗ : J → F and a map (cid:101) V : (cid:101) R → (cid:101) Σ c which (see [38] Theorem 4.1) extends to a corresponding CT -mapˆ i V : Λ J → Λ F . Lemma 5.13. If ˆ i ∞ ( ξ ) / ∈ P ∞ , then ( ρ n , ξ ) satisfies EP ( ξ ) .Proof. Translating by an appropriate element of G we may assume without loss of generalitythat the geodesic ray [0 , ξ ) is contained in (cid:101) R so that ξ ∈ Λ J . Write ¯ ξ = ˆ i V ( ξ ). Since as above¯ ρ n satisfies U EP P , then certainly ( ¯ ρ n , ¯ ξ ) satisfies EP ( ¯ ξ ). Hence there is a strictly increasingfunction f : N → N such that if x ∈ [ O, ¯ ξ ) and d ( O, x ) > N then d H ( (cid:101) ψ n ( x ) , O ) > f ( N ). Now given N ∈ N and ¯ x ∈ [ O, ¯ ξ ), consider the H -geodesic segment λ = [ O, (cid:101) ψ n (¯ x )] from O to (cid:101) ψ n (¯ x ), and let H ( λ ) denote the collection of horoballs traversed by λ . Let P N (¯ x, n ) bethe total length of the geodesic segments in [ O, (cid:101) ψ n (¯ x )] ∩ H \ H ( λ ) and Q N (¯ x, n ) = |H ( λ ) | bethe number of horoballs traversed by λ . We claim there exist a strictly increasing function g : N → N and M N ∈ N such that(5) P N (¯ x, n ) + Q N (¯ x, n ) ≥ g ( N ) for all ¯ x ∈ [ O, ¯ ξ ) , ¯ x / ∈ B H ( O, N ) and n ≥ M N . If the claim is false, then there exists
K > N , there exist ¯ x N ∈ [ O, ¯ ξ ),¯ x N / ∈ B H ( O, N ) and arbitrarily large n ∈ N such that(6) P N (¯ x N , n N ) + Q N (¯ x N , n N ) ≤ K. Inductively, choose n = n N > n N − . Then (6) implies in particular that there is a uniformbound to the number of horoballs traversed by the ray [ O, (cid:101) ψ n N (¯ x N )]. By slightly adjustingconstants, we can assume that each horoball is penetrated to a distance at least a > a . Now by hypothesis d ( O, (cid:101) ψ n N (¯ x N )) ≥ f ( N ). Thus there can be no uniform upperbound to the distance travelled through each horoball, in other words, we can find a sequence H N of horoballs such that the length of the segment (cid:101) ψ n N (¯ x N ) ∩ H N tends to infinity with N .By choosing the first such horoball traversed, we can assume that H N intersects B H ( O, K ).Passing to a subsequence if necessary, we may assume that all horoballs H N are based at thepoint ˆ k n ( η ) for some fixed parabolic point η ∈ Λ F . Thus we can find a sequence y N ∈ [ O, ¯ x N ]such that (cid:101) ψ n N ( y N ) ∈ H N and d ( O, (cid:101) ψ n N ( y N )) → ∞ . Since the rays (cid:101) ψ n N ([ O, ¯ ξ )) converge to (cid:101) ψ ∞ ([ O, ¯ ξ )) uniformly on compact subsets in H (by U EP P for the sequence ¯ ρ n ), this meansthat (cid:101) ψ n N ( y N )) → ˆ j ∞ ( η ). On the other hand, (cid:101) ψ n N ( y N ) is arbitrarily close in the Euclideanmetric on B ∪ ˆ C to ˆ k n N ( η ) for large N . Hence ˆ k ∞ ( ¯ ξ ) = ˆ k ∞ ( η ).Since η is a parabolic point in Λ F , by Bowditch’s Theorem 5.2 this means that either¯ ξ ∈ V ∗ ( P ) or ¯ ξ is the endpoint of a leaf in the crown of the unstable lamination of χ . Sinceˆ i V : Λ J → Λ F is one-to-one except on P , the same is true of ξ . Since by assumption ξ / ∈ P , wededuce that ξ is the end of a boundary leaf of the crown of χ , which gives, using Theorem 5.2again, ˆ i ∞ ( ξ ) ∈ P ∞ , contrary to hypothesis. This proves claim (5).Now we will show that claim (5) implies that ( ρ n , ξ ) satisfies EP ( ξ ). As above, let D n = B n ∪ T n and let (cid:101) D n denote the lift of D n corresponding to (cid:101) R above, that is, whose stabiliseris ρ n ( J ). Let (cid:101) B n be the corresponding lift of B n . The map V induces an obvious uniformlybi-Lipschitz map V n : B n = X R × [0 , n ] → E cn = Σ c × [0 , n ], where E n = Σ × [0 , n ] is the modelof the convex core of Q ( χ n (Σ) , Σ) as in Section 5.1.6. Clearly V n ◦ φ n = ψ n ◦ V , while on thelevel of fundamental groups, ( V n ) ∗ ◦ ρ n = ¯ ρ n ◦ V ∗ and ( V n ) ∗ , V ∗ are group isomorphisms. IMITS OF LIMIT SETS II 39
Since Margulis tubes are convex, it follows as in [20] that (cid:101) D n satisfies the condition ofLemma 2.6 relative to the collection T n of lifts of T n it contains, as does (cid:101) E n relative to the setof horoballs H n say. Let (cid:101) D en , (cid:101) E en denote the corresponding electric spaces. Note that V ∗ inducesa bijective correspondence between T n and H n .To avoid having to define the extension of (cid:101) V n to the whole of (cid:101) D en we proceed as follows.Suppose that λ is an electric quasi-geodesic in (cid:101) D en with endpoints in (cid:101) B n . Replace λ with apath ˆ λ which runs along the boundaries of the electrocuted sets in T n as follows. Suppose somesegment λ (cid:48) of λ enters and leaves some T ∈ T n at points a, b respectively. Replace λ (cid:48) by thesegment ( a, [0 , ∪ ( b, [0 , ⊂ ∂T × [0 ,
1] of electric length 2. Since the sets in T are uniformlyseparated, the resulting path ˆ λ is still an electric quasi-geodesic. Now extend the definition of (cid:101) V n to a map, still denoted (cid:101) V n , which sends ∂T × [0 , → ∂H × [0 ,
1] in the obvious way, where H ∈ H corresponds to T ∈ T . Using the fact that V n is uniformly bi-Lipschitz, it is easy to seethat (cid:101) V n (ˆ λ ) is an electric quasi-geodesic in (cid:101) E cn , and that the number of electrocuted componentstraversed by ˆ λ and (cid:101) V n (ˆ λ ) is the same.Now suppose ξ ∈ Λ J as in the statement of the Lemma. Since ξ / ∈ P the ray [ O, ξ ) iscontained in the convex hull of (cid:101) X R ⊂ (cid:101) X . Note however that the path (cid:101) V ([ O, ξ )) may not be aquasi-geodesic in (cid:101)
Σ as it is contained in Σ c and thus may skirt round the boundaries of horoballsin Σ. We can nevertheless work with the ray (cid:101) V ([ O, ξ )), which ends at the point ˆ i V ( ξ ) = ¯ ξ ,see for example [38] Theorem 4.1. (We obtain a quasi-geodesic ray from (cid:101) V ([ O, ξ )) by replacingeach segment which skirts a horoball with the corresponding geodesic joining the entry and exitpoints, see for example [38] especially Lemma A.5.)Let x ∈ [ O, ξ ) and let ¯ x = (cid:101) V ( x ). We want to compare the ray [ O, (cid:101) φ n ( x )] ⊂ (cid:101) K n to the ray[ O, (cid:101) ψ n (¯ x )] ⊂ (cid:101) E n . Since (cid:101) D n is quasi-convex in (cid:101) K n , we can after bounded adjustments assumethat [ O, (cid:101) φ n ( x )] ⊂ (cid:101) D n . Since (cid:101) V n (cid:101) φ n = (cid:101) ψ n (cid:101) V we have (cid:101) V n ( (cid:101) φ n ( x )) = (cid:101) ψ n (¯ x ).Replacing the electric geodesic λ say from O to (cid:101) φ n ( x ) in (cid:101) D en by the corresponding electricquasi-geodesic ˆ λ as above, we see that (cid:101) V n (ˆ λ ) is a well-defined electric quasi-geodesic in (cid:101) E en withendpoint (cid:101) ψ n (¯ x ). Moreover (cid:101) V n (ˆ λ ) has length comparable to ˆ λ . Since (5) effectively says that thelength of (cid:101) V n (ˆ λ ) in the electric metric on (cid:101) E en goes to infinity uniformly with N independently of n , the same is true of ˆ λ . This proves that ( ρ n , ξ ) satisfies EP ( ξ ) and we are done. (cid:3) Corollary 5.14. If ˆ i ∞ ( ξ ) / ∈ P ∞ , then ( ρ n , ξ ) satisfies EP P ( ξ ) .Proof. Continuing with the notation of Lemma 5.13, let λ be a geodesic segment in [ O, ξ ) outside B H ( O, N ) and let ¯ λ = (cid:101) V ( λ ). As in the previous lemma, note that ¯ λ may not be a geodesic asit is contained in Σ c and thus may skirt round the boundary of a horoball in Σ. This leads to anannoying technical issue in that it is convenient only to work with segments ¯ λ whose endpoints lie outside the lifts of the horoball Σ \ Σ c . To fix this, note that in Definition 5.8 of condition EP P ( ξ ) for convergence, it is clearly enough to check the condition for an increasing sequenceof values N < N < . . . . Since (cid:101) V ([ O, ξ )) does not terminate in the cusp, we may thereforerestrict to those N i for which the first x ∈ (cid:101) V ([ O, ξ )) outside B ( O, N i ) is outside a horoball.Thus given ¯ λ as above, by extending forwards and backwards along (cid:101) V ([ O, ξ )) if necessary, wemay assume that its initial and final points are outside horoballs in (cid:101)
Σ.Now consider the geodesic [ (cid:101) ψ n (¯ λ )] in (cid:101) E en and let µ, µ ea be respectively the electric geodesicand the electro-ambient geodesic with the same endpoints, as in Section 2.6. By Lemma 2.6, µ ea is a bounded distance from [ (cid:101) ψ n (¯ λ )]. Using EP P ( ¯ ξ ), we deduce that µ ea is outside B H ( O, g ( N ) − k ), for some uniform k >
0. Then using the same method as in the previous lemma, it followsthat any point on µ is outside some ball B ( O, h ( N )) in the electric metric on (cid:101) E en for somefunction h ( N ) → ∞ with N .Now using the same trick as in the previous lemma, replace µ with the electric quasi-geodesic ˆ µ and apply the map (cid:101) V − n . We obtain an electric quasi-geodesic ˆ ν = (cid:101) V − n (ˆ µ ) in (cid:101) D en with the same endpoints as (cid:101) φ n ( λ ). Since (cid:101) V − n is bi-Lipschitz with respect to electric metrics, forany point Q ∈ ˆ ν we have d e ( O, Q ) (cid:31) h ( N ), where we write X (cid:31) Y to mean there is a uniformconstant c > X > cY .By Lemma 2.6 again, it will be enough to show that the electro-ambient quasi-geodesicobtained from ˆ ν by replacing each segment which cuts through an equidistant tube T ∈ T n with the hyperbolic geodesic with the same endpoints, is outside some ball B H ( O, f ( N )) forsome function f ( N ) → ∞ with N .Suppose that A and B are the entry and exit points of ˆ ν to some T ∈ T n . Suppose thatthe hyperbolic geodesic from to O to A first meets T at a point ¯ A . Since d e ( O, A ) (cid:31) h ( N ), itfollows that d H ( O, ¯ A ) (cid:31) h ( N ). We deduce from Lemma 5.15 below that T is entirely outside B ( O, R ) for some R (cid:31) h ( N ). In particular the hyperbolic geodesic segment [ A, B ] is outside B ( O, R ) and the result follows. (cid:3)
Lemma 5.15.
Suppose that T is an equidistant tube in H , that is, the set of points equidistantfrom a geodesic axis in H , and that T has radius at least r for some uniformly large r . Supposethat A ∈ ∂T is outside B ( O, R ) , where O ∈ H is a fixed base-point. Then the entire tube T isoutside B ( O, R (cid:48) ) for some R (cid:48) (cid:31) R .Proof. Let P be the point on T nearest to O in the hyperbolic metric and let P (cid:48) , A (cid:48) be thefeet of the perpendiculars from P, A to the axis of T . We claim that if d ( A, P ) > c >
1, thenthe angle θ between the geodesics [ A, P ] and [
A, A (cid:48) ] is uniformly bounded away from π/
2. If d ( A (cid:48) , P (cid:48) ) >
1, this is easy since [
A, P ] roughly tracks the quasi-geodesic [
A, A (cid:48) ] ∪ [ A (cid:48) , P (cid:48) ] ∪ [ P (cid:48) , P ]. IMITS OF LIMIT SETS II 41 If d ( A (cid:48) , P (cid:48) ) ≤ Q, Q (cid:48) be respectively the feet of the perpendiculars from A (cid:48) , P (cid:48) to thegeodesic [ A, P ], so that | QQ (cid:48) | <
1. Since cos θ = tanh | AQ | / tanh | AA (cid:48) | and | AA (cid:48) | ≥ r , it followsthat cos θ is bounded away from 0 unless | AQ | is very small. By symmetry | AP | = 2 | AQ | + | QQ (cid:48) | ,so if we assume that | AP | > c > O, A ] is outside T , and since [ A, A (cid:48) ]is perpendicular to ∂T at A , we have shown that either [ A, P ] has uniformly bounded length,or the angle between [
O, A ] and [
A, P ] is bounded away from 0.It follows in all cases that [
O, A ] ∪ [ A, P ] is a uniform quasi-geodesic and hence that d ( O, P ) (cid:31) R . The result follows by convexity. (cid:3) Corollary 5.16. If ˆ i ∞ ( ξ ) / ∈ P ∞ , then ˆ i n ( ξ ) → ˆ i ∞ ( ξ ) .Proof. This follows immediately from Theorem 5.9. (cid:3)
This completes the proof of Proposition 5.12.
Remark 5.17.
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