Boundaries of Dehn fillings
BBOUNDARIES OF DEHN FILLINGS
DANIEL GROVES, JASON FOX MANNING, AND ALESSANDRO SISTO
Abstract.
We begin an investigation into the behavior of Bowditch and Gro-mov boundaries under the operation of Dehn filling. In particular we showmany Dehn fillings of a toral relatively hyperbolic group with 2–sphere bound-ary are hyperbolic with 2–sphere boundary. As an application, we show thatthe Cannon conjecture implies a relatively hyperbolic version of the Cannonconjecture.
Contents
1. Introduction 12. Preliminaries 53. Weak Gromov–Hausdorff convergence 124. Spiderwebs 155. Approximating the boundary of a Dehn filling 206. Proofs of approximation theorems for hyperbolic fillings 237. Approximating boundaries are spheres 398. Ruling out the Sierpinski carpet 429. Proof of Theorem 1.2 4410. Proof of Corollary 1.4 45Appendix A. δ –hyperbolic technicalities 46References 501. Introduction
One of the central problems in geometric group theory and low-dimensionaltopology is the Cannon Conjecture (see [Can91, Conjecture 11.34], [CS98, Con-jecture 5.1]), which states that a hyperbolic group whose (Gromov) boundary is a2-sphere is virtually a Kleinian group. By a result of Bowditch [Bow98] hyperbolicgroups can be characterized in terms of topological properties of their action onthe boundary. The Cannon Conjecture is that (in case the boundary is S ) thistopological action is in fact conjugate to an action by M¨obius transformations. Rel-atively hyperbolic groups are a natural generalization of hyperbolic groups whichare intended (among other things) to generalize the situation of the fundamentalgroup of a finite-volume hyperbolic n -manifold acting on H n .A relatively hyperbolic group pair ( G, P ) has associated with it a natural com-pact space ∂ ( G, P ) called the Bowditch boundary [Bow12, §
9] on which it acts asa geometrically finite convergence group, so that every parabolic fixed point hasstabilizer conjugate to a unique element of P . The motivating example is when a r X i v : . [ m a t h . G R ] D ec DANIEL GROVES, JASON FOX MANNING, AND ALESSANDRO SISTO
G < SO ( n,
1) is a geometrically finite Kleinian group and P a collection of con-jugacy representatives of maximal parabolic subgroups. In this case the Bowditchboundary coincides with the limit set. A result of Yaman [Yam04] characterizesrelatively hyperbolic groups in terms of their action on the Bowditch boundary.It is natural to wonder whether a relatively hyperbolic group whose Bowditchboundary is a 2-sphere is virtually Kleinian. In fact, both the Cannon Conjectureand this relative version are special cases of a much more general conjecture ofMartin and Skora [MS89, Conjecture 6.1]. One of the main results of this paper (seeCorollary 1.4 below) is to prove that the relative version of the Cannon Conjecturefollows from the absolute version.If the peripheral subgroups P of a relatively hyperbolic group pair ( G, P ) arethemselves hyperbolic, then so is G , and it therefore acts as a uniform convergencegroup on its Gromov boundary ∂G . (The relationship between these boundaries isexplained in [Tra13], see also [Ger12, GP13, MOY12, Man15].) In [Osi07, GM08](cf. [DGO]), the operation of group theoretic Dehn filling is developed, and is shownto satisfy a coarse analog of Thurston’s Hyperbolic Dehn Surgery Theorem [Thu80,Section 5.8]. This is to say that many “Dehn fillings” of a relatively hyperbolicgroup pair are themselves relatively hyperbolic.Relatively hyperbolic Dehn filling has found many important applications, in-cluding in the proof of the virtual Haken conjecture [Ago13] and the solution of theisomorphism problem in a large class of relatively hyperbolic groups [DG15].In the classical setting one begins with a relatively hyperbolic group pair ( G, P )whose Bowditch boundary is a 2–sphere and ends with a hyperbolic group G whoseGromov boundary is again a 2–sphere. On the other hand, examples of CAT( − G whose boundary is much more complicatedthan that of ( G, P ), but which nonetheless admits a fairly explicit description. Onepurpose of this paper is to begin an investigation of whether these results are specialto fillings of manifolds, or are reflective of more general phenomena. To this end, weobtain a description (Theorems 5.2, 5.3 and 5.4) of the boundary of a Dehn fillingas a certain kind of limit of quotients of subsets of the original boundary by discretegroups. The following result is contained in Theorems 5.2 and 5.3 (see Definition3.1 for the definition of weak Gromov-Hausdorff convergence). For simplicity, westate it in the case of one peripheral subgroup, that is P = { P } . Theorem 1.1.
Let G = G/ (cid:104)(cid:104) N (cid:105)(cid:105) be a sufficiently long hyperbolic filling of therelatively hyperbolic pair ( G, { P } ) , with N (cid:47) P infinite. Then there is a sequenceof Gromov hyperbolic spaces X i whose boundaries ∂X i weakly Gromov-Hausdorffconverge to ∂G , if we endow all these boundaries with suitable metrics. Moreoverthere is an exhaustion K < K < · · · of ker( G → G ) so that each ∂X i can beidentified with (( ∂ ( G, { P } ) \ Λ( K i )) /K i ) ∪ F , where F is a union of finitely many copies of ∂ ( P/N ) . This gives a new way to prove statements about boundaries of Dehn fillings, byproving a statement about the approximating ∂X i , and showing it persists in thelimit. Theorem 5.4 states that under some additional assumptions, we may assumeall these metrics are uniformly linearly connected, which helps control the limit. OUNDARIES OF DEHN FILLINGS 3
Our main application is the following statement, which says roughly that suffi-ciently long Dehn fillings of relatively hyperbolic groups with 2–sphere boundarymust have 2–sphere boundary.
Theorem 1.2.
Let G be a group, and P = { P , . . . , P n } a collection of free abeliansubgroups. Suppose that ( G, P ) is relatively hyperbolic, and that ∂ ( G, P ) is a –sphere.Then for all sufficiently long fillings G → G = G ( N , . . . , N n ) with P i /N i virtu-ally infinite cyclic for each i , we have that G is hyperbolic with ∂G homeomorphicto S . Note that if ∂ ( G, P ) is a 2–sphere, then any parabolic acts properly cocompactlyon R . If we are assuming it is free abelian, it must therefore be Z . If we didn’tassume abelian, we might have to worry about higher genus surface groups asperipheral groups. These higher genus surface groups being hyperbolic groups,we should exclude them from the peripheral structure to get a boundary whichis a Sierpinski carpet. Conjecturally, a hyperbolic group with Sierpinski carpetboundary is virtually Kleinian. Kapovich and Kleiner [KK00] prove that this wouldfollow from the Cannon Conjecture.One can make a relative version of the Cannon Conjecture as follows (cf. [Kap07,Problem 57]): Conjecture 1.3. (Relative Cannon Conjecture) Let ( G, P ) relatively hyperbolicgroup with ∂ ( G, P ) ∼ = S and all elements of P free abelian. Then G is Kleinian. We remark that the usual Cannon conjecture says ‘virtually Kleinian’ because anon-elementary hyperbolic group may not act faithfully on its boundary; there maybe a finite kernel. However, under the assumption that the parabolic subgroups ofa nonelementary relatively hyperbolic group (with nontrivial peripheral structure)are free abelian, there are no nontrivial finite normal subgroups, and so ‘Kleinian’is the expected conclusion.We have the following corollary of Theorem 1.2; see Section 10 for the proof.
Corollary 1.4.
The Cannon Conjecture implies the Relative Cannon Conjecture.
This resolves [Kap07, Problem 60], though we do not proceed via Kapovich’ssuggested method of proof.1.1.
Sketch proof of Theorem 1.2.
We must somehow reconstruct ∂G frominformation about ∂ ( G, P ). It is a result of Dahmani–Guirardel–Osin that K =ker( G → G ) is (for a sufficiently long filling) freely generated by parabolic sub-groups [DGO]. Associated to ( G, P ) is a proper, Gromov hyperbolic space (the combinatorial cusped space ) X = X ( G, P ) on which G acts geometrically finitely(cocompactly away from horoballs); the Gromov boundary of this space is equiv-ariantly homeomorphic to ∂ ( G, P ).In Section 4 we develop an analog in the cusped space of the “windmills” tech-nology of [DGO] to obtain an exhaustion of K by free products of finitely many parabolic subgroups. Our replacements for windmills are called spiderwebs – theseform an exhaustion W ⊂ W · · · of the cusped space by quasiconvex subsets, eachof which is acted on geometrically finitely by a finitely generated subgroup K n of K . The “partial quotients” X/K n approximate a cusped space X = X/K for therelatively hyperbolic pair ( G, P ). But in the situation of interest G is itself hy-perbolic, so we need approximations to ∂G , not to ∂ ( G, P ). Such approximations DANIEL GROVES, JASON FOX MANNING, AND ALESSANDRO SISTO are obtained from
X/K n by removing finitely many images of (deep) horoballs of X . We must take some care to ensure that these truncated partial quotients areuniformly hyperbolic over all n . With even more care, we are able to show theseboundaries are uniformly linearly connected over all n (Theorem 5.4), and thatthey have nice descriptions in terms of ∂ ( G, P ) (Theorem 5.2).Once this is ensured, we have a sequence of spaces which converge in the pointedGromov–Hausdorff topology to a G –cocompact space. Their boundaries thereforeconverge (in a sense described in Section 3) to the boundary of G (Theorem 5.3).The above results apply more generally when G is hyperbolic, and the result ofa long filling of a relatively hyperbolic pair ( G, P ), and in fact we state versions inthe setting of the Bowditch boundary of ∂ ( G, P ) as Theorems 5.5–5.7. The proofsof these relative versions are strictly easier than those of Theorem 5.2–5.4, thoughwe do not provide the relative proofs in this paper.In Section 7 we specialize to ∂ ( G, P ) ∼ = S , and P i /K i virtually cyclic. Inthis case we can show the approximating boundaries are spheres by a homologicalargument.We now sketch the argument that the boundary is planar. Results from [GM]show that the boundary is a Peano continuum without local cut points. We theninvoke a characterization of Claytor [Cla34], which says that a Peano continuumwithout cut points is planar if and only if it contains no non-planar graph. Anadaptation of a lemma of Ivanov (Lemma 3.9) shows that if ∂G contained such agraph, then so would all but finitely many of the approximating boundaries. Sincethey are spheres, they do not.Since ∂G is planar, connected, and has no local cut points, a result of Kapovichand Kleiner [KK00, Theorem 4] implies that it is either S or a Sierpinski Carpet.In Subsection 8 we rule out the Sierpinski Carpet.1.2. Outline.
Section 2 contains background, notation and preliminary results;the reader can skim it and refer back to it when needed.In Section 3 we introduce the notion of weak Gromov–Hausdorff convergence,which plays an important role in our description of the boundary of a Dehn filledgroup.In Section 4 we introduce spiderwebs, a variation of the windmills from [DGO].We cannot use windmills directly for our purposes, but our construction is verysimilar to that in [DGO].Finally, the main contributions of this paper start with Sections 5 and 6, where westate and prove our main results about general Dehn filling. As discussed above, wedescribe the boundary of a Dehn filled group as a certain weak Gromov–Hausdorfflimit of spaces, each the boundary of a certain hyperbolic space, whose topologywe have control on.Starting with Section 7, we focus on the setup of Theorem 1.2, that is to saywe consider fillings of a relatively hyperbolic pair whose Bowditch boundary isa 2–sphere. First of all, we exploit the general description of the approximatingboundaries to show that in that situation they are all spheres. meaning a connected, locally connected, compact metrizable space OUNDARIES OF DEHN FILLINGS 5
Section 8 contains the last missing piece of the proof of Theorem 1.2: Weprove that a weak Gromov–Hausdorff limit of simply connected spaces is (cid:15) -simply-connected for every (cid:15) >
0, therefore proving that the Sierpinski carpet cannot be alimit of spheres.In Section 9 we prove Theorem 1.2, which at that point only requires puttingtogether various pieces.In Section 10, we prove Corollary 1.4, which requires arguments about limits ofrepresentations in Isom( H ).Finally, in Appendix A we record some technical results which are surely wellknown to experts but for which we do not know of a reference in the literature.1.3. Acknowledgments.
The authors would like to thank Peter Ha¨ıssinsky andGenevieve Walsh for useful conversations, and an anonymous referee for severalhelpful comments.This material is based upon work supported by the National Science Foundationunder grant No. DMS-1440140 while the second and third authors were in residenceat the Mathematical Sciences Research Institute in Berkeley, California, during theFall 2016 semester. The first author is partially supported by a grant from theSimons Foundation (
Preliminaries
For a point p of a metric space ( M, d ), write S R ( p ) for { x ∈ M | d ( x, p ) = R } ,and B R ( p ) for { x ∈ M | d ( x, p ) ≤ R } . If M is a geodesic space we write [ x, y ] for achoice of geodesic from x to y in M .In a geodesic space ( Z, d ), every geodesic triangle ∆ comes with a surjective mapto a possibly degenerate comparison tripod T ∆ , which is isometric on each side ofthe triangle, and so the vertices map to feet of the tripod. If the vertices of thetriangle are x , y , and z , the leg corresponding to x has length( y | z ) x : = 12 ( d ( y, x ) + d ( z, x ) − d ( y, z )) , also known as the Gromov product of y and z with respect to x .For δ >
0, the geodesic space Z is a δ –hyperbolic space if all geodesic trianglesin Z are δ –thin , in the sense that the map to the comparison tripod has fibers ofdiameter at most δ . A space is Gromov hyperbolic if it is δ –hyperbolic for some δ .See [BH99, III.H] for more details, and the relationship with other definitions.A Gromov hyperbolic space Z has a boundary at infinity or Gromov boundary ∂Z , which can be defined in terms of sequences of points. Namely, a sequence { x i } converges to infinity if lim i,j →∞ ( x i | x j ) p = ∞ for some (or equivalently every)basepoint p . Two sequences { x i } and { y i } are equivalent if lim i,j →∞ ( x i | y j ) p = ∞ ,and ∂Z is defined to be the set of equivalence classes of sequences which convergeto infinity. If the equivalence class of { x i } is ξ , we write { x i } → ξ . In a properGromov hyperbolic space, ∂Z can also be defined as equivalence classes of geodesicrays, where two rays are counted as equivalent if they have images which are finiteHausdorff distance apart. All the spaces we consider are proper. DANIEL GROVES, JASON FOX MANNING, AND ALESSANDRO SISTO
For p ∈ Z and ξ, υ ∈ Z ∪ ∂Z , the Gromov product is extended as follows:( ξ | υ ) p = sup (cid:26) lim inf i,j →∞ ( x i | y j ) p (cid:12)(cid:12)(cid:12)(cid:12) { x i } → ξ, { y i } → υ (cid:27) . It is a standard fact, see e.g. [V¨ai05, Lemma 5.6] or [BH99, III.H.3.17.(5)],that, up to a small error, one can compute the Gromov product using any givenrepresentative sequences, meaning that if { x i } → ξ, { y i } → υ then lim inf( x i | y j ) p is within 2 δ of ( ξ | υ ) p .The following observation is [BH99, III.H.3.17.(3)]: Lemma 2.1.
Let Z be Gromov hyperbolic, and let p ∈ Z . For any ξ, υ in Z ∪ ∂Z ,there are sequences { x i } → ξ , and { y i } → υ so that lim n →∞ ( x n | y n ) p = ( ξ | υ ) p . We also consider the Gromov product of geodesic rays ( α | β ) p with respect totheir common starting point p , which we define to be( α | β ) p = lim inf s,t →∞ ( α ( s ) | β ( t )) p . Visual metrics on the boundary of a Gromov hyperbolic space.
Forany given parameter (cid:15) > p ∈ X , the function ( η, ξ ) (cid:55)→ e − (cid:15) ( η | ξ ) p behaves somewhat like a metric on ∂X , though it may not satisfy the triangleinequality. It does makes sense to ask whether e − (cid:15) ( ·|· ) p is bilipschitz or quasi-isometric to some metric on ∂X .We recall the definition: Definition 2.2.
Let Z be a Gromov hyperbolic space, with basepoint w . A vi-sual metric on ∂Z , based at w , with parameters (cid:15), κ is a metric ρ ( · , · ) which is κ –bilipschitz to e − (cid:15) ( ·|· ) w .From [BH99, III.H.3.21] one can fairly readily deduce the following: Proposition 2.3. [BH99, III.H.3.21]
Let δ > . Then for all positive (cid:15) ≤ δ thereis a κ = κ ( (cid:15), δ ) ≥ with lim (cid:15) → κ ( (cid:15), δ ) = 1 so that:If Z is a δ –hyperbolic space and p ∈ Z , then ∂Z has a visual metric based at p with parameters (cid:15), κ . Visual metrics are hardly ever length metrics, and in fact hardly ever admit recti-fiable paths. However, the notion of linear connectedness is a useful “replacement”for the notion of length metric.
Definition 2.4.
Let L ≥
1. A metric space M is L –linearly connected if everypair of points x, y ∈ M is contained in a connected subset J of diameter at most L · d ( x, y ). We say M is linearly connected if it is L –linearly connected for some L . Remark 2.5.
As observed, for example, in the introduction of [Mac08], if M iscompact then up to increasing L by an arbitrarily small amount we can assumethat J is an arc. We frequently make this assumption in the rest of the paper.A homeomorphism f : X → Y of metric spaces is a quasi-symmetry if there is ahomeomorphism η : [0 , ∞ ) → [0 , ∞ ) so that d ( f ( x ) , f ( y )) d ( f ( x ) , f ( z )) ≤ η (cid:18) d ( x, y ) d ( x, z ) (cid:19) OUNDARIES OF DEHN FILLINGS 7 for all triples of distinct points x, y, z ∈ X . The spaces X and Y are then said tobe quasi-symmetric . All visual metrics on the boundary of a given hyperbolic spaceare quasi-symmetric to each other. Observe: Lemma 2.6. If X is linearly connected, then so is any space quasi-symmetric to X . The cusped space associated to a relatively hyperbolic pair.
In thissection we associate a metric graph (the (combinatorial) cusped space ) to a relativelyhyperbolic pair, and fix notation for various subsets of it.
Definition 2.7.
Let Γ be a graph, endowed with the metric that gives each edgelength 1. The combinatorial horoball based on
Γ is the metric graph H (Γ) whosevertex set is Γ (0) × Z ≥ , and with two types of edges:(1) A vertical edge of length 1 from ( v, n ) to ( v, n + 1) for any v ∈ Γ (0) andany n ≥ k >
0, if v and w are vertices of Γ so that 0 < d Γ ( v, w ) ≤ k then thereis a single horizontal edge of length 1 joining ( v, k ) to ( w, k ).Define the depth of a vertex D ( v, n ) = n and extend the depth function affinelyover edges.The inverse image D − ( n ) for n an integer is called the horosphere at depth n .This is a graph whose vertices are in bijection with those of Γ. The distance in D − ( n ) between two vertices ( v, n ) and ( w, n ) is (cid:100) − n d Γ ( v, w ) (cid:101) .If H = H (Γ) for some Γ, and I is a nondegenerate interval in R , we define H I = D − ( I ).Let ( G, P ) be a group pair (so G is a group and P is a collection of subgroups),and suppose that G and the elements of P are all finitely generated. Choose agenerating set S for G which contains a generating set for each P ∈ P . (This iscalled a compatible generating set.) Let Γ be the Cayley graph for G with respectto S , metrized so each edge has length 1. Each left coset gP of P ∈ P spans aconnected gP g − –invariant subgraph Γ( gP ) ⊂ Γ. Definition 2.8.
The cusped space X ( G, P ) is obtained from Γ by attaching, foreach P ∈ P , and each coset gP , a copy of H ( gP ), by identifying Γ( gP ) to thehorosphere at depth 0 of H ( gP ).The cusped space is not quite determined by the pair ( G, P ), since we had tochoose a generating set, but any two choices give quasi-isometric spaces, by [Gro13,Corollary 6.7]. Definition 2.9. ( G, P ) is relatively hyperbolic if and only if the cusped space X ( G, P ) is Gromov hyperbolic.In [GM08, Theorem 3.25] it is proved that this definition is equivalent to otherdefinitions of relative hyperbolicity, in the finitely generated case. See [Hru10] foran extension of this definition to the non-finitely generated case. Throughout thispaper, we are only concerned with the case that G and all elements of P are finitelygenerated. We recall the following useful property of horoballs in the cusped spaceof a relatively hyperbolic group. Lemma 2.10. [GM08, Lemma 3.26]
Suppose X = X ( G, P ) is δ –hyperbolic, andthat H ⊂ X is a combinatorial horoball. For any integer R ≥ δ , the set H [ R, ∞ ) isconvex in X . DANIEL GROVES, JASON FOX MANNING, AND ALESSANDRO SISTO
Definition 2.11.
Suppose that ( G, P ) is relatively hyperbolic, and suppose thateach element of P is infinite. Let X ( G, P ) be the associated cusped space. TheGromov boundary ∂X ( G, P ) is called the Bowditch boundary of ( G, P ).In case some elements of P are finite, then ∂X ( G, P ) contains isolated points.If P ∞ is the collection of infinite elements of P , then ( G, P ∞ ) is also relativelyhyperbolic, and its Bowditch boundary can be obtained from ∂X ( G, P ) by removingthe isolated points.In case G itself is hyperbolic, Bowditch characterized which ( G, P ) are relativelyhyperbolic. Recall a family of subgroups P is almost malnormal if whenever P ∩ gP g − is infinite, for P , P ∈ P and g ∈ G , we have P = P and g ∈ P . Theorem 2.12. [Bow12, Theorem 7.11]
Let G be hyperbolic, and suppose P is afamily of distinct subgroups of G . The pair ( G, P ) is relatively hyperbolic if andonly if P is an almost malnormal family of quasi-isometrically embedded subgroups. Dehn fillings.Definition 2.13.
Let G be a group and let P = { P , . . . , P n } be a finite collectionof subgroups of G . Given a collection of normal subgroups N i (cid:69) P i , called fillingkernels , the quotient G → G ( N , . . . , N n ) = G/K , where K = (cid:104)(cid:104) (cid:83) i N i (cid:105)(cid:105) , is calleda (Dehn) filling of ( G, P ). We say that a property holds for all sufficiently longfillings of ( G, P ) if there is a finite set B ⊆ G \ { } so that whenever N i ∩ B = ∅ for all i , the group G/K has the property.
Theorem 2.14. [Osi07, GM08]
Let ( G, P = { P , . . . , P n } ) be relatively hyperbolic.Then for any finite subset F ⊆ G \ { } the following holds. For any sufficientlylong filling φ : G → G/K we have(1) for each i , φ induces an embedding of P i /N i in G/K whose image we iden-tify with P i /N i ,(2) ( G/K, { P /N , . . . , P n /N n } ) is relatively hyperbolic,(3) φ restricted to F is injective. For any relatively hyperbolic pair ( G, P ), the peripheral groups P always consistof an almost malnormal family of quasi-isometrically embedded subgroups [Osi06,Proposition 2.36 and Lemma 5.4]. Hence we have the following corollary of Theorem2.14. Corollary 2.15.
Let ( G, P ) be relatively hyperbolic. For all sufficiently long fillings G → G/K , the filling
G/K is hyperbolic if and only if every P i /N i is hyperbolic. The following is an easy consequence of the third part of Theorem 2.14.
Lemma 2.16.
Let ( G, P ) be relatively hyperbolic, with associated cusped space X .Then for any R ≥ the following holds. For any sufficiently long filling G → G/K the restriction of the map X → X/K to any ball of radius R centered at an elementof the Cayley graph is an isometry onto its image. Moreover, the same holds truefor the map X → X/K where K < K is any subgroup. The next result is proved in [AGM09] assuming that G is torsion-free, but thisassumption is not necessary, as explained in the proof of [Ago13, Theorem A.43].Alternatively, it follows from Lemma 2.16 and the Coarse Cartan–Hadamard The-orem (Theorem 6.2 below). OUNDARIES OF DEHN FILLINGS 9
Proposition 2.17. [AGM09, Proposition 2.3]
Suppose that ( G, P ) is relativelyhyperbolic, and fix a generating set for G as in Definition 2.8. There exists a δ sothat (i) the cusped space for ( G, P ) is δ –hyperbolic; and (ii) For all sufficiently longfillings ( G, P ) → ( G, P ) the cusped space for ( G, P ) (with respect to the image ofthe fixed generating set for G ) is δ –hyperbolic. Geometry of truncated horoballs.
In the classical 2 π Theorem of Gromovand Thurston, a cusped hyperbolic 3–manifold is modified to a closed negativelycurved one by replacing each cusp neighborhood by a thick “Margulis tube” arounda short geodesic [BH96]. In the universal cover this tube lifts to a large neighbor-hood of a geodesic line.In our setting we model our Dehn filled group G = G/K by a space which canbe either thought of as(1) The quotient cusped space
X/K , with certain deep horoballs removed, or(2) The Cayley Graph of G , with certain truncated horoballs added.The truncated horoballs are analogous to the neighborhoods of geodesic lines dis-cussed above. The same space with truncated horoballs omitted would still beGromov hyperbolic, but we would lose control of various constants and be unableto make uniform statements over all long fillings.Let θ > θ -hyperbolic Cayley graph. It follows (see[BH99, § III.H.1.22]) that Γ satisfies Gromov’s 4-point condition Q ( θ ): for all x, y, z, w ∈ Γ, d ( x, w ) + d ( y, z ) ≤ max { d ( x, y ) + d ( z, w ) , d ( x, z ) + d ( y, w ) } + 2 θ. Definition 2.18.
Let t (Γ) be the smallest integer so that the graph H t (Γ) satisfiesthe Gromov 4-point condition Q (5); we argue below that this is well-defined.Let H (Γ) be the combinatorial horoball based on Γ. As noted in Section 2, themetric on vertices in H k = D − ( k ) is defined by d H k ( v, w ) = (cid:100) − k d Γ ( v, w ) (cid:101) . This formula and the defining equation for Gromov products shows that t (Γ) existsand t (Γ) ≈ log ( θ ). By the proofs of [BH99, Propositions III.H.1.17, III.H.1.22]this implies that triangles in H t (Γ) are 30-thin. The graph H t (Γ) is a 30–hyperbolicCayley graph. The loops of length at most 481 based at a vertex give the relationsin a Dehn presentation (see the proof of [BH99, III.Γ.2.6]). Attaching disks toall loops of length at most 481 in H t (Γ) , we therefore obtain a simply connectedcomplex with linear combinatorial isoperimetric function with constant 1.In [GM08, §
3] a simply-connected 2-complex is built from H (Γ) by attachingvertical squares and pentagons and horizontal triangles, and the depth function D is extended across these 2-cells. Let (cid:101) H (Γ) be this simply-connected 2-complex,and denote the extended depth function by (cid:101) D . For I an interval in R , define (cid:101) H I (Γ) = (cid:101) D − ( I ). Then H I (Γ) (defined in above) is the 1-skeleton of (cid:101) H I (Γ).The space (cid:101) H (Γ) satisfies a linear combinatorial isoperimetric function with con-stant 3, by [GM08, Proposition 3.7]. The proof of this can be easily adapted byfilling at depth t (Γ) using the disks from the Dehn presentation, to prove the fol-lowing result. Proposition 2.19.
Suppose that Γ is a θ -hyperbolic Cayley graph, and that t (Γ) is chosen as in Definition 2.18. The –complex (cid:101) H [0 ,t (Γ)] (Γ) satisfies a linear com-binatorial isoperimetric inequality with constant . Since we also have a universal bound on the length of boundaries of disks, [GM08,Proposition 2.23] gives the following.
Corollary 2.20.
Let θ > and suppose that Γ is a θ -hyperbolic Cayley graph. Thegraph H [0 ,t (Γ)] (Γ) is θ –hyperbolic, for a universal constant θ . It is straightforward to see that if 0 ≤ a ≤ t (Γ) then H [ a,t (Γ)] (Γ) is convex in H [0 ,t (Γ)] (Γ). Therefore we have the following result. Corollary 2.21.
Let θ > and suppose that Γ is a θ -hyperbolic Cayley graph. Forany a ∈ [0 , t (Γ)] , the graph H [ a,t (Γ)] (Γ) is θ –hyperbolic, for the same constant θ from Corollary 2.20. It is possible to understand geodesics in H [ a,t (Γ)] (Γ) in a very similar way togeodesics in H (Γ). The following result can be be proved using almost exactly thesame proof as [GM08, Lemma 3.10]. Lemma 2.22.
Let θ > and suppose that Γ is a θ -hyperbolic Cayley graph. Let a ∈ [0 , t (Γ)] and suppose that p, q ∈ H [ a,t (Γ)] (Γ) are distinct vertices. There is ageodesic γ in H [ a,t (Γ)] (Γ) between p and q which consists of at most two verticalsegments and a single horizontal segment. Moreover, if this horizontal segment isnot at depth t (Γ) then it has length at most . The next lemma tells us that truncated horoballs are “locally visual”.
Lemma 2.23.
Let θ > and suppose that Γ is a θ -hyperbolic Cayley graph. Supposethat t (Γ) > λ > θ .For any a ∈ [ λ, t (Γ)] and any p, q ∈ H [ a,t (Γ)] (Γ) so that d ( p, q ) = λ there exists ageodesic [ p, q (cid:48) ] of length λ in H [ a − λ,t (Γ)] (Γ) so that d ( q, [ p, q (cid:48) ]) ≤ θ .Proof. We assume that p, q are vertices.Choose a geodesic [ p, q ] of the form as in the conclusion of Lemma 2.22. Thereare a number of possibilities.If either [ p, q ] has two vertical segments, or p has greater depth than q , then[ p, q ] ends at q with a vertical segment heading towards the horosphere H ∼ = Γ. Inthis case, append a vertical segment of length λ + 10 θ to [ p, q ] to form a new path σ of length 2 λ + 10 θ . It is straightforward to see that σ is a 10 θ –local geodesic,and so σ lies within 2 θ of any geodesic between the endpoints of σ (see [BH99,III.H.1.13]). Taking an initial subpath of length 2 λ of any such geodesic gives apath [ p, q (cid:48) ] as in the conclusion of the lemma.Suppose next that q has greater depth than p and that there is a horizontalsegment of length at least 3 at the end of [ p, q ]. In this case, appending a verticalpath of length λ + 10 θ from q to the end of [ p, q ] creates a 10 θ –local geodesic, andwe proceed as in the first case.The only remaining case is that q has greater depth than p and that [ p, q ] isentirely vertical or terminates with a horizontal segment of length at most 2. Let d be the depth of p . We claim that there exists y ∈ H d at distance at least 2 λ − from p . In fact, if this was not the case then H d + λ − would be contained in a 1–ballaround some point (that lies vertically below p ), and hence H d + λ − would have OUNDARIES OF DEHN FILLINGS 11 diameter 1, implying that t (Γ) is at most d + λ −
3. However, the depth of q is atleast d + λ −
2, a contradiction. There is a H [ a − λ,t (Γ)] (Γ)–geodesic from p to y thatgoes straight down from p distance ≥ λ −
3, along a short horizontal segment andthen straight up to y . The path [ p, q ] lies in the 5–neighborhood of this geodesic,which can be prolonged to a geodesic of length 2 λ , still contained in H [ a − λ,t (Γ)] (Γ),if needed. (cid:3) Finally, we prove that for sufficiently long fillings, the value of t (Γ) (the partialtruncation depth above) can also be made large. This follows quickly from thefollowing straightforward result. Lemma 2.24.
Suppose that H is a group with finite generating set S . For any A > there exists B so that for any nontrivial normal subgroup J of H so that (i) H/J is a hyperbolic group; and (ii) J contains no nontrivial elements of length lessthan B (with respect to the word metric d S ), the Cayley graph of H/J with respectto the image of S is not A –hyperbolic.Proof. Let h be the shortest nontrivial element of J . Consider a geodesic γ in theCayley graph of H from 1 to h , which gives a loop p in the Cayley graph of H/J .It is easily seen that any subgeodesic of γ of length at most half the word length | h | S gives a geodesic in the Cayley graph of H/J . In particular, the loop p canbe subdivided into a geodesic bigon where the midpoint of one side is at distance | h | S / (cid:3) Suppose that ( G, P ) is relatively hyperbolic. Fix a compatible finite generatingset S . Suppose that G ( N , . . . , N m ) is a Dehn filling with each P i /N i being hy-perbolic. According to Corollary 2.20 there are constants t ( i ) so that the partiallytruncated horoballs of the Cayley graphs of P i /N i to depth t ( i ) is θ –hyperbolic, fora universal constant θ . Moreover, t ( i ) depends only on the hyperbolicity constantof the Cayley graph of P i /N i (with respect to the obvious generating set in theimage of S ). Moreover, from the construction and Lemma 2.24, it is clear that thishyperbolicity constant goes to infinity and so does t ( i ). Therefore, the following isan immediate consequence of Lemma 2.24. Corollary 2.25.
Let
C > be any constant. For all sufficiently long fillings G ( N , . . . , N m ) of ( G, P ) , the constants t ( i ) are all larger than C . A Greendlinger Lemma.
Roughly speaking, the next lemma says that, for G → G/K a sufficiently long filling, if we have some x ∈ G and g ∈ K \ { } , thenany geodesic [ x, gx ] in the cusped space for G goes deep into a horoball, and it canbe shortened using an element of the conjugate of the filling kernel corresponding tothe horoball. It is similar to [DGO, Lemma 5.10] and it can presumably be provenusing the techniques we use in Section 4 to construct spiderwebs, but we give asimple proof that only relies on Proposition 2.17 and Lemma 2.16.For a relatively hyperbolic pair ( G, P ) with cusped space X , let C denote the col-lection of parabolic points for the G –action on ∂X . Suppose that G → G ( N , . . . , N m )is a Dehn filling, with N i (cid:69) P i . The points in C are limit points of horoballs of theform H ( gP i ) for g ∈ G . If c ∈ C is of the form c = ∂ H ( gP i ) then let K c = gN i g − .We also write H c for H ( gP i ). Lemma 2.26.
Let ( G, P ) be relatively hyperbolic, with associated cusped space X .For any D ≥ , for any sufficiently long filling G → G/K the following holds: Forany g ∈ K \ { } and x ∈ X there exists c ∈ C so that (1) any geodesic [ x, gx ] intersects H Dc ,(2) for any geodesic [ x, gx ] there exists k ∈ K c so that d ( x, kgx ) < d ( x, gx ) .Proof. As in Proposition 2.17, let δ ≥ • X is δ –hyperbolic, and • for all sufficiently long fillings X/K is δ –hyperbolic.Fix D ≥
0. It follows from Lemma 2.16 that for any sufficiently long filling G → G/K , and any x, y ∈ X in the same K –orbit satisfying d ( x, y ) ≤ δ , thereexists a horoball H c so that x, y ∈ H [ D, + ∞ ) c . Hence y = kx for some k ∈ K c sincethe intersection of K and the stabilizer of H c is K c .Let K be the kernel of such a filling, and let g ∈ K \ { } and x ∈ X . If d ( x, gx ) ≤ δ then we are done by the argument above, so suppose d ( x, gx ) > δ .Let γ = [ x, gx ] be a geodesic in X from x to gx and let ˜ γ be the projected pathin X/K . Since ˜ γ is a loop of length at least 10 δ , ˜ γ is not a 10 δ –local geodesic.Therefore, there are two points p and q appearing in this order along γ with d ( p, q ) ≤ δ and some k ∈ K so that d ( p, kq ) < d ( p, q ). By the argument above, we have p, q ∈ H [ D, + ∞ ) c for some horoball H c , and hence γ ∩ H Dc (cid:54) = ∅ . Also, we have k ∈ K c and d ( x, kgx ) ≤ d ( x, p ) + d ( p, kq ) + d ( kq, kgx ) < d ( x, p ) + d ( p, q ) + d ( q, gx ) = d ( x, gx ) , as required. (cid:3) Weak Gromov–Hausdorff convergence
Our strategy to describe boundaries of Dehn fillings involves describing them aslimits, in a suitable sense, of metric spaces that we have more control over. Thecorrect notion of limit for our purposes is similar to that of Gromov–Hausdorff limitand is described as follows.
Definition 3.1.
Let ( M i , d i ) i ∈ N and ( M, d ) be metric spaces. We say that (
M, d )is a weak Gromov–Hausdorff limit of the sequence ( M i , d i ) if there exists λ ≥ λ, (cid:15) i )–quasi-isometries M → M i , with (cid:15) i → i → ∞ . Example 3.2.
If the compact metric space (
M, d ) is a weak Gromov–Hausdorfflimit of the sequence of connected metric spaces ( M i , d i ), then ( M, d ) is connected.In fact, if M is not connected then we can write M = A (cid:116) B with A, B non-emptyand d ( A, B ) = (cid:15) >
0. It is readily seen that for n large enough M n inherits a similardecomposition and hence it is not connected.This section has two goals. The first one is to show that when a sequence ofhyperbolic spaces converges in a suitable sense to a hyperbolic space, then theirboundaries weakly Gromov–Hausdorff converge to the boundary of the limit hy-perbolic space. The second goal is to give a criterion which allows us to prove(using a result of Claytor [Cla34]) that the weak Gromov–Hausdorff limit of a se-quence of metric spaces homeomorphic to S is planar. The criterion we prove inthis section, Lemma 3.9, is an adaptation of a result of Ivanov [Iva97].3.1. From convergence of spaces to convergence of their boundaries.
Thedefinition of convergence of hyperbolic spaces that we use is the following one.
OUNDARIES OF DEHN FILLINGS 13
Definition 3.3.
Let (
X, p ) be a pointed metric space. Say the sequence of pointedmetric spaces { ( X i , p i ) } i ∈ N strongly converges to ( X, p ) if the following holds: Forevery
R >
0, there are isometries φ i : B R ( p ) → B R ( p i ) with φ i ( p ) = p i for all butfinitely many i .In order to relate the boundary of a hyperbolic space to spheres of large radius,we need the space to be D –visual in the following sense – a different concept fromthat of a visual metric given in Definition 2.2. Definition 3.4.
Let
D >
0. A geodesic metric space X is D –visual if, for every a, b ∈ X , there is a geodesic ray based at a passing within D of b .The following is the main result of this subsection and it is an immediate corollaryof Lemma 3.8 below. Proposition 3.5.
Let δ > . Suppose { ( X i , p i ) } i ∈ N strongly converges to ( X, p ) ,and that the spaces X and X i are all δ –hyperbolic and δ –visual. Then for all positive (cid:15) ≤ δ and κ as in Proposition 2.3, and any visual metrics ρ i on ∂X i , ρ on ∂X with parameters (cid:15), κ , the boundary ( ∂X, ρ ) is a weak Gromov–Hausdorff limit of ( ∂X i , ρ i ) . Remark 3.6.
With a bit more work it should be possible to weaken the assump-tion of strong convergence in 3.5 to the assumption of pointed Gromov–Hausdorffconvergence.
Lemma 3.7.
Let X be δ –hyperbolic and let w ∈ X . Let α, β be rays starting at w with limit points a, b ∈ ∂X . Let T be the tripod obtained by gluing two rays togetheralong an initial subsegment of length ( a | b ) w . Then there is a (1 , δ ) –quasi-isometryfrom α ∪ β to T , isometric on each of α and β .Proof. Let s, t ∈ [0 , ∞ ). We must show that d ( α ( s ) , β ( t )) is within 5 δ of the distanceof their images in T : τ ( s, t ) = (cid:40) | s − t | min { s, t } ≤ ( a | b ) w s + t − a | b ) w otherwiseBy Lemma 2.1, there are sequences { a i } → a and { b i } → b with lim i →∞ ( a i | b i ) w =( a | b ) w . Choose n, N so that ( a n | α ( N )) w and ( b n | β ( N )) w both exceed max { s, t } +10 δ , and so that ( a n | b n ) w is within η ≤ δ of ( a | b ) w . Let α (cid:48) be a geodesic from w to a n , and let β (cid:48) be a geodesic from w to b n . We have d ( α ( s ) , α (cid:48) ( s )) and d ( β ( s ) , β (cid:48) ( s ))both bounded above by δ .Suppose first that one of s, t is at most ( a | b ) w . Without loss of generality it is s . Then d ( α (cid:48) ( s ) , β (cid:48) ( s )) ≤ δ + 2 η ≤ δ . It follows that d ( α ( s ) , β ( s )) ≤ δ , and so d ( α ( s ) , β ( t )) is within 4 δ of τ ( s, t ) = | s − t | .Finally suppose that both s and t are larger than ( a | b ) w . Consider a geodesictriangle ∆ two of whose sides are α (cid:48) and β (cid:48) . Let a and b be the images in thecomparison tripod for ∆ of α (cid:48) ( s ) and β (cid:48) ( t ), respectively. Then the distance d ( a, b ) = s + t − a n | b n ) w is within 2 η of τ ( s, t ) = s + t − a | b ) w . Thus d ( α (cid:48) ( s ) , β (cid:48) ( t )) differsfrom τ ( s, t ) by at most 2 δ + 2 η , and d ( α ( s ) , β ( t )) differs from τ ( s, t ) by at most4 δ + 2 η ≤ δ . (cid:3) For the next lemma, recall (as we observed in Section 2.1) that even though e − (cid:15) ( ·|· ) may not be a metric, the concept of quasi-isometry still makes sense. Also, recall that given a point p of a metric space ( M, d ), we denote the sphere of radius R around p , that is to say the set { x ∈ M | d ( x, p ) = R } , by S R ( p ). Lemma 3.8.
For every δ, (cid:15) there exists λ so that the following holds. Let X be δ –hyperbolic and δ –visual. Then for any w ∈ X and any R > , there is a ( λ, c ) –quasi-isometry φ : ( ∂X, e − (cid:15) ( ·|· ) w ) → ( S R ( w ) , e − (cid:15) ( ·|· ) w ) , where c = c ( δ, (cid:15), R ) tends to as R tends to + ∞ .Proof. All rays in this proof are rays starting at w . Denote e − (cid:15) ( ·|· ) by ρ ( · , · ). We’llprove the lemma for λ = e (cid:15)δ and c = e (cid:15) ( δ − R ) . Note that c tends to 0 as R tendsto ∞ .Let us define a map φ : ∂X → S R ( p ). For a ∈ ∂X , choose a ray γ a (parametrizedby arc length) representing it. Then, set φ ( a ) = γ a ( R ).The fact that X is δ –visual combined with the fact that asymptotic rays staywithin distance δ of each other implies that for any x ∈ S R ( w ) there exists a ∈ ∂X with, say, d ( x, φ ( a )) ≤ δ , and hence ρ ( x, φ ( a )) ≤ e − (cid:15) ( R − δ ) < c . Hence, the imageof φ is c –dense in S R ( w ).Now let a, b ∈ ∂X . We distinguish two cases.If ( a | b ) w ≥ R then ρ ( a, b ) ≤ e − (cid:15)R . In this case φ ( a ) and φ ( b ) lie within 5 δ ofeach other by Lemma 3.7. In particular ρ ( φ ( a ) , φ ( b )) ≤ e (cid:15)δ − (cid:15)R , so the difference | ρ ( φ ( a ) , φ ( b )) − ρ ( a, b ) | is at most e (cid:15)δ − (cid:15)R = c. Suppose on the other hand ( a | b ) w ≤ R , so that ρ ( a, b ) ≥ e − (cid:15)R . By Lemma 3.7, | d ( φ ( a ) , φ ( b )) − R − ( a | b ) w ) | ≤ δ , and hence | ( φ ( a ) | φ ( b )) w − ( a | b ) w | ≤ δ . Wededuce λ − ρ ( a, b ) ≤ ρ ( φ ( a ) , φ ( b )) ≤ λρ ( a, b ) , with no additive error in this case. (cid:3) Linear connectedness and a lemma of Ivanov.
In order to show that theboundary of our filled group is planar in the proof of Theorem 1.2 in Section 9, weuse the following adaptation of a lemma of Ivanov [Iva97, Lemma 2.2].
Lemma 3.9.
Let ( M i , d i ) be metric spaces, and assume that each M i is (homeo-morphic to) a closed smooth manifold of dimension ≥ . Suppose that there exists L so that each M i is L -linearly connected and that ( M, d ) is a weak Gromov–Hausdorfflimit of ( M i , d i ) . If the finite graph Γ can be topologically embedded in M then forall large enough i it can also be embedded in M i . We emphasize that we do not assume that the limit M is a manifold. Proof.
By assumption there is some K ≥
1, and a sequence of (
K, (cid:15) ( i ))–quasi-isometries π i : M → M i with (cid:15) ( i ) → i → + ∞ .Let f : Γ → M be a topological embedding. We fix some constants C, (cid:15), (cid:15) (cid:48) satisfying:(1)
C > K L ;(2) for any disjoint subgraphs Γ , Γ of Γ we have d ( f (Γ ) , f (Γ )) ≥ C(cid:15) ;(3) if the edges e , e share the endpoint v and p i ∈ e i is so that d ( f ( p i ) , f ( v )) ≥ (cid:15)/C then d ( f ( p ) , f ( p )) ≥ C(cid:15) (cid:48) ; and(4) (cid:15) (cid:48) < (cid:15) KL .Fix i so that (cid:15) ( i ) ≤ (cid:15) (cid:48) until the end of the proof. For v a vertex of Γ, let ˜ v = π i ( f ( v )). OUNDARIES OF DEHN FILLINGS 15
Claim 3.9.1.
We can choose, for each vertex v of Γ, a path-connected neighborhood U ˜ v of ˜ v so that B (cid:15) (˜ v ) ⊆ U ˜ v ⊆ B L(cid:15) (˜ v ). Moreover, we can require U ˜ v to be a compactmanifold (with boundary). Proof of Claim 3.9.1.
For x, y ∈ M i , let A x,y be the union of all paths of length ≤ Ld i ( x, y ) joining x to y . Let A = (cid:83) { A x,y | x, y ∈ B (cid:15) (˜ v ) } . Notice that A ⊆ B (2 L +1) (cid:15) (˜ v ), and that A is path-connected. Fix a homeomorphism h from a smoothmanifold to M i , and let g : M i → [0 , ∞ ) be chosen so that g ◦ h is smooth and g − (0)is the closure of A .For any R > (2 L + 1) (cid:15) , and any sufficiently small regular value t of g ◦ h , we have g − ([0 , t ]) ⊆ B R (˜ v ). In particular, we can fix t so that U t = g − ([0 , t ]) ⊆ B L(cid:15) (˜ v ).We may take U ˜ v to be the connected component of U t containing A . (cid:3) Claim 3.9.2.
We can choose, for each edge e of Γ, an embedded path γ e in M i insuch a way that the following properties are satisfied. If v is not an endpoint of e then γ e does not intersect U ˜ v , and it intersects U ˜ v exactly in one endpoint if v isan endpoint of e . Moreover, the paths γ e are disjoint. Proof of Claim 3.9.2.
Let e be an edge of Γ, and let { p j } j =1 ,...,n be a sequence ofpoints along f ( e ) that subdivide f ( e ) into subpaths of diameter ≤ (cid:15) (cid:48) . For each j set q j = π i ( p j ). Consider paths γ j connecting q j to q j +1 of diameter ≤ Ld i ( q j , q j +1 ).Let A e be the union of all such paths, and notice that A e ⊂ N L ( K +1) (cid:15) (cid:48) ( π i ( f ( e ))).Suppose v is not an endpoint of e . We claim A e ∩ U ˜ v is empty. Indeed, d ( f ( v ) , f ( e )) ≥ C(cid:15) , so d i (˜ v, A e ) ≥ CK (cid:15) − (cid:15) i − L ( K + 1) (cid:15) (cid:48) ≥ (cid:18) KL − K (cid:19) (cid:15) > L(cid:15).
Suppose e and e (cid:48) are edges of Γ not sharing an endpoint. We claim A e ∩ A e (cid:48) isempty. Indeed, d i ( A e , A e (cid:48) ) ≥ d i ( π i ( e ) , π i ( e (cid:48) )) − L ( K + 1) (cid:15) (cid:48) ≥ CK (cid:15) − (cid:15) i − L ( K + 1) (cid:15) (cid:48) > . Finally suppose that e and e (cid:48) are edges which do share an endpoint v . We claim A e ∩ A e (cid:48) ⊆ ˚ U ˜ v . Indeed, if x ∈ A e ∩ A e (cid:48) , there are q j ∈ π i ( f ( e )) and q (cid:48) k ∈ π i ( f ( e (cid:48) ))within L ( K + 1) (cid:15) (cid:48) of x . The corresponding points p j ∈ f ( e ) and p (cid:48) k ∈ f ( e (cid:48) ) mustsatisfy d ( p j , p (cid:48) k ) ≤ K ( L ( K + 1) (cid:15) (cid:48) + (cid:15) (cid:48) ) < C(cid:15) (cid:48) . Using the condition (3), it followsthat d ( p j , f ( v )) is bounded above by (cid:15)C , and so d ( q j , ˜ v ) ≤ KC (cid:15) + (cid:15) i . Finally d ( x, ˜ v ) ≤ KC (cid:15) + (cid:15) i + L ( K + 1) (cid:15) (cid:48) < (cid:15) .It is now easy to see that the path-connected set A e contains an embedded path γ e as required. (cid:3) In order to conclude the construction we just need to observe that, since U ˜ v is amanifold of dimension at least 2, each U ˜ v contains a union of paths P ˜ v that pairwiseonly intersect at ˜ v , each connecting ˜ v to an endpoint of some γ e .The union (cid:83) P ˜ v ∪ (cid:83) γ e is a subset of M i homeomorphic to Γ. (cid:3) Spiderwebs
In this section we make a construction similar to that of windmills from [DGO, § spiderwebs . The main difference between the con-structions is that we want the stabilizers of spiderwebs to be a free product of finitely many factors, which is not the case for the windmills from [DGO]. We work in this section with a θ –hyperbolic space, reserving the symbol δ fora constant that is chosen in later sections (and depends on θ ). We will will fix aparticular θ in Assumption 6.1 and then fix δ = 1500 θ in Assumption 6.4.4.1. Notation.
We fix the following notation from now until the end of the section.Let ( G, P ) be a relatively hyperbolic pair, and let X be a cusped space for the pairas in Definition 2.8. Fix an arbitrary integer θ ≥ X is θ –hyperbolic. As inSection 2.5, let C be the collection of parabolic fixed points in ∂X . We are going tochoose a G –equivariant, 10 θ –separated family of horoballs as follows: Let c ∈ C .Then c is the unique limit point of some H c = H ( gP ), for some coset gP of some P ∈ P . Let (cid:98) H c = H [500 θ, ∞ ) c ; this is convex in X by Lemma 2.10. Note that theclosure of the complement of (cid:83) (cid:98) H c is G –cocompact.Suppose that { N i (cid:67) P i } is a collection of (long) filling kernels, with N i (cid:54) = { } . Asin Section 2.5, if c = ∂ H ( gP i ), then let K c = gN i g − be the conjugate of a fillingkernel fixing c . We suppose that the groups K c satisfy the following: Very translating condition.
For each c ∈ C , g ∈ K c \ { } and x ∈ X \ (cid:98) H c wehave d X ( x, gx ) ≥ θ .The following is an easy consequence of Theorem 2.14. Lemma 4.1.
For sufficiently long fillings the family { K c } satisfies the very trans-lating condition. Spiderwebs.
For Y a subset of X we denote C ( Y ) = { c ∈ C | Y ∩ (cid:98) H c (cid:54) = ∅} . Definition 4.2 (Spiderweb) . A θ –spiderweb is a subset W of X containing 1 andsatisfying the following axioms.(S1) W is 4 θ –quasiconvex.(S2) C ( W ) = C ( N θ ( W )).(S3) The group K W generated by (cid:91) c ∈C ( W ) K c , preserves W . Moreover, for any R >
0, ( N R ( G ) ∩ W ) /K W is compact.(S4) There exists a finite subset C ⊂ C ( W ) so that K W is the free product ∗ c ∈ C K c .Here is the main theorem of this section: Theorem 4.3.
In the notation established in Subsection 4.1, and for K = (cid:104)(cid:104) (cid:83) i N i (cid:105)(cid:105) ,there exists a family of θ –spiderwebs W ⊂ W · · · so that (cid:83) W i = X and (conse-quently) K = (cid:83) K W i . To extend a given θ –spiderweb W to a larger one, we need a few lemmas abouthow W interacts with its translates under elements of K c for (cid:98) H c near to W , butnot intersecting W . Define C (cid:48) ( W ) to be C ( N θ ( W )) \ C ( W ). Lemma 4.4.
Let c ∈ C (cid:48) ( W ) . Then diam X ( π (cid:98) H c ( W )) ≤ θ and diam X ( π W ( (cid:98) H c )) ≤ θ .Proof. Note that (cid:98) H c is convex, and W is 4 θ –quasiconvex. Moreover, d ( (cid:98) H c , W ) ≥ θ by property ((S2)) of θ –spiderwebs. The lemma then follows by applyingLemma A.12. (cid:3) OUNDARIES OF DEHN FILLINGS 17
Lemma 4.5.
Suppose c ∈ C (cid:48) ( W ) , and g ∈ K c \ { } . Let γ be a geodesic joining W to gW .(1) The geodesic γ intersects (cid:98) H c in a subsegment of length at least θ .(2) The geodesic γ is contained in N θ ( W ) ∪ N θ ( (cid:98) H c ) ∪ N θ ( gW ) .Proof. Let γ be a geodesic joining w ∈ W to gw (cid:48) ∈ gW .Note that gπ (cid:98) H c ( W ) = π (cid:98) H c ( gW ). Lemma 4.4 says π (cid:98) H c ( W ) has diameter at most8 θ . By the very translating condition d X ( π (cid:98) H c ( W ) , π (cid:98) H c ( gW )) is at least (10 − θ .In particular d X ( π (cid:98) H c ( w ) , π (cid:98) H c ( gw (cid:48) )) > θ . Using the second part of Lemma A.12,the geodesic γ passes within 6 θ of both π (cid:98) H c ( W ) and π (cid:98) H c ( gW ). In particular thereare points p, p (cid:48) on γ at depth at least (500 − θ in the horoball containing (cid:98) H c , andsatisfying d X ( p, p (cid:48) ) ≥ (10 − − θ . Since geodesics in combinatorial horoballsare vertical except for up to three horizontal edges (Lemma 2.22), γ must intersect (cid:98) H c in a subsegment of length at least (10 − − − θ − > θ , establishingthe first claim of the Lemma.Turning to the second claim, let σ be a shortest geodesic joining W to (cid:98) H c , andlet σ be a shortest geodesic from (cid:98) H c to gW . Note that each of σ , σ has length atmost 100 θ . Lemma A.13 implies that the part of γ between w and (cid:98) H c is containedin N θ ( W ∪ (cid:98) H c ) ∪ N θ ( σ ). Similarly the part of γ between (cid:98) H c and w (cid:48) is containedin N θ ( (cid:98) H c ∪ gW ) ∪ N θ ( σ ). Thus γ ⊆ N θ ( W ∪ gW ) ∪ N θ ( (cid:98) H c ) , as required. (cid:3) Lemma 4.6.
Let c, c (cid:48) ∈ C (cid:48) ( W ) be distinct, and let g ∈ K c \ { } , g (cid:48) ∈ K c (cid:48) \ { } .Then d X ( π W ( gW ) , π W ( g (cid:48) W )) ≥ θ .Proof. By way of contradiction, suppose that g ∈ K c \ { } and g (cid:48) ∈ K c (cid:48) \ { } satisfy d X ( π W ( gW ) , π W ( g (cid:48) W )) < θ .We claim that π W ( gW ) ⊆ π W ( (cid:98) H c ) and π W ( g (cid:48) W ) ⊆ π W ( (cid:98) H c (cid:48) ). Indeed, suppose x ∈ π W ( gW ). Then there is some y ∈ gW with d ( y, W ) = d ( y, x ). By Lemma4.5.(1), any geodesic from x to y intersects (cid:98) H c . Let z ∈ (cid:98) H c be on one such geodesic.Then d ( z, W ) = d ( z, x ), so x ∈ π W ( z ) ⊆ π W ( (cid:98) H c ). This establishes that π W ( gW ) ⊆ π W ( (cid:98) H c ); the argument that π W ( g (cid:48) W ) ⊆ π W ( (cid:98) H c (cid:48) ) is identical.Thus we also have d X ( π W ( (cid:98) H c ) , π W ( (cid:98) H c (cid:48) )) < θ . By Lemma 4.4, these projec-tions have diameter at most 16 θ . Since (cid:98) H c and (cid:98) H c (cid:48) are each distance at most 100 θ from W , we deduce d X ( (cid:98) H c , (cid:98) H c (cid:48) ) < θ + 2(16 θ ) + 2(100 θ ) = 732 θ < θ . Sincethe horoballs are 10 θ –separated, this contradicts c (cid:54) = c (cid:48) . (cid:3) Since { } is a θ –spiderweb, Theorem 4.3 follows immediately from the followingproposition. Proposition 4.7.
Suppose that W is a θ –spiderweb. Then there is a θ –spiderweb W (cid:48) so that(1) W (cid:48) contains N θ ( W ) ,(2) K W (cid:48) = K W ∗ (cid:18) ∗ c ∈ E K c (cid:19) for some finite subset E ⊆ C ( W (cid:48) ) \ C ( W ) .Proof. If C ( N θ ( W )) = C ( W ) then W (cid:48) = N θ ( W ) is a θ –spiderweb, and the othercondition trivially holds. Therefore, suppose that C (cid:48) ( W ) = C ( N θ ( W )) \ C ( W ) is nonempty. Note that C (cid:48) ( W ) has finitely many K W –orbits, because of item (S3) in the definition of θ –spiderweb.Let E be a set of representatives for the K W –orbits of C (cid:48) ( W ), let K + W = (cid:104) K W ∪ ( (cid:83) c ∈ E K c ) (cid:105) and let W (cid:48) be the union of all geodesics connecting pairs of points in theorbit K + W N θ W . By Lemma A.10, W (cid:48) is 2 θ –quasiconvex. We remark that Lemma4.5.(1) (together with non-triviality of the N i ) implies that C (cid:48) ( W ) ⊆ C ( W (cid:48) ).Our goal is now to prove that W (cid:48) is a θ –spiderweb and K + W = K W (cid:48) .Let φ : K W ∗ (cid:18) ∗ c ∈ E K c (cid:19) → K + W be the natural map. Clearly φ is surjective. We establish ((S4)) in the definition of θ –spiderweb by showing that φ is injective. At the same time, we obtain informationabout geodesics between K + W –translates of W sufficient to establish ((S2)) in thedefinition of a θ –spiderweb. Claim.
Let w, w (cid:48) ∈ W , and let g ∈ K W ∗ (cid:18) ∗ c ∈ E K c (cid:19) . Let γ be a geodesic joining w to φ ( g ) w (cid:48) . Let H = (cid:83) c ∈ E (cid:98) H c .(1) The geodesic γ lies in N θ ( K + W W ) ∪ N θ ( K + W H ).(2) Let x ∈ γ \ N θ ( K + W W ). Then the distance from x to K + W W is at mostdepth( x ) − θ .(3) If g / ∈ K W , then φ ( g ) w (cid:48) / ∈ W .We complete the proof of the Proposition, assuming the claim. Axiom ((S1)),quasiconvexity, follows from Lemma A.10, as already noted.We next show that Axiom ((S2)) holds. In fact, we show K + W ( C ( W ) ∪ E ) = C ( W (cid:48) ) = C ( N δ ( W (cid:48) )). The containments “ ⊆ ” are clear, so we are left to show ifsome horoball (cid:98) H c satisfies d X ( (cid:98) H c , W (cid:48) ) ≤ θ then c ∈ K + W ( C ( W ) ∪ E ). Let x ∈ W (cid:48) minimize the distance to (cid:98) H c . The point x is on some geodesic joining points in K + W .N θ ( W ). It therefore lies within 12 θ of a geodesic joining points in K + W .W .Translating everything by an element of K + W , we may assume that this geodesic hasone endpoint in W , as in the claim. Part (1) of the claim implies that x lies eitherin a 46 θ –neighborhood of some K + W –translate of W , or in a 142 θ –neighborhood ofsome K + W –translate k. (cid:98) H c (cid:48) of (cid:98) H c (cid:48) for some c (cid:48) ∈ E . In the first case, we concludethat (cid:98) H c has a K + W –translate meeting a 100 θ –neighborhood of W , implying that c ∈ K + W ( C ( W ) ∪ E ). In the second case, we have d ( (cid:98) H c , k. (cid:98) H c (cid:48) ) ≤ θ , implying c = kc (cid:48) by 10 θ –separation of horoballs, and hence c ∈ K + W ( E ).The invariance of W (cid:48) under K + W is immediate from the construction. Also, K + W = K W (cid:48) because C ( W (cid:48) ) = K + W ( C ( W ) ∪ E ), so we get the first part of Axiom((S3)). The second part of Axiom ((S3)) follows from part (2) of the Claim.Since φ is automatically injective on K W , part (3) of the claim shows φ is injec-tive, establishing Axiom ((S4)), and showing W (cid:48) is a θ –spiderweb. Proof of Claim.
We’ll prove the claims by building a nice 100 θ –local 12 θ –tightquasigeodesic joining w to gw (cid:48) , and applying quasigeodesic stability. Since g lies ina free product, it can be written g = g · · · g n where each g i is a nontrivial elementof some free factor. Without any loss of generality, we may assume that g (cid:54) = 1, and OUNDARIES OF DEHN FILLINGS 19 (rechoosing w (cid:48) if necessary) that g n / ∈ K W . We define certain prefixes k i = g · · · g j i of g inductively as follows: j = (cid:40) g / ∈ K W g ∈ K W , and j i +1 = (cid:40) j i + 1 g j i +1 / ∈ K W j i + 2 g j i +1 ∈ K W Thus, for example, k = g if g / ∈ K W and k = g g otherwise. We obtainelements k , . . . , k s , where k s = g . These choices ensure that, if we define W = W and W i = k i W for i ∈ { , . . . , s } , we always have W i (cid:54) = W i − . For each i ∈{ , . . . , s } choose a shortest geodesic [ q i − , p i ] from W i − to W i . This is a translateof a segment joining W to κW for some κ ∈ K c \ { } , c ∈ C (cid:48) ( W ). Lemma 4.5.(1)implies that [ q i − , p i ] has length at least 100 θ .Note that q i − ∈ π W i − ( p i ) and p i ∈ π W i ( q i − ).Set p = w , and q s = φ ( g ) w (cid:48) . For each i ∈ { , . . . s } choose a geodesic [ p i , q i ];this geodesic lies in a 4 θ –neighborhood of W i by quasiconvexity. When i / ∈ { , s } ,we have p i ∈ π W i ( W i − ) and q i ∈ π W i ( W i +1 ), so by Lemma 4.6, it has length atleast 500 θ .Let α be the broken geodesic [ p , q ] · [ q , p ] · · · [ p s , q s ]. We claim that α isa 100 θ –local 12 θ –tight path. Except possibly for the first and last segments, allthe geodesic subsegments of α have length at least 100 θ , so tightness need onlybe verified on concatenations of two of the geodesic subsegments. One of thesesegments always connects a point to a closest point projection in some W i whichcontains both endpoints of the second segment. Since W i is 4 θ –quasiconvex, wecan apply Lemma A.15 to conclude that this concatenation of two subsegments is12 θ –tight.We can now apply Lemma A.16, with C = 12 θ , to conclude that any geodesic γ with the same endpoints as α is Hausdorff distance at most 28 θ from α . Inparticular, such a geodesic α does not lie in a 4 θ –neighborhood of W , so φ ( g ) w (cid:48) / ∈ W and part (3) of the claim is established.To establish part (1), we note that α lies in N θ ( K + W W ) ∪ N θ ( K + W H ) byapplying Lemma 4.5.(2) to the subsegments passing between the W i . It followsthat any geodesic from w to φ ( g ) w (cid:48) lies in N θ ( K + W W ) ∪ N θ ( K + W H ).To establish part (2), let x lie on γ . Then x lies within 28 θ of some point x (cid:48) on α . If x (cid:48) ∈ [ p i , q i ] for some i , then d X ( x, K + W W ) ≤ θ . Otherwise, x (cid:48) ∈ [ q i , p i ],which is entirely contained in the 100 θ –neighborhood of some (cid:98) H c . In particular,the depth of x (cid:48) is at least 400 θ + d ( x, K + W W ) (cid:3) This completes the proof of Proposition 4.7. (cid:3)
We have already noted that Theorem 4.3 follows immediately from Proposition4.7, so we have proved Theorem 4.3 and completed the construction of θ –spiderwebs.The following result follows immediately from the construction of spiderwebsand may be useful in future applications. Theorem 4.8.
Suppose that ( G, P ) is a relatively hyperbolic pair, let X be thecusped space for ( G, P ) and let C be the set of parabolic fixed points in ∂X .For all sufficiently long fillings, the following holds: Let K be the kernel of thefilling. There is a set T ⊂ C meeting each K –orbit exactly once, so that K = ∗ t ∈ T ( K ∩ Stab( t )) , and each subgroup K ∩ Stab( t ) is conjugate in G to a unique filling kernel N i (cid:67) P i . Proof.
Fix a long enough filling G → G ( N , . . . , N m )so that the very translating condition above holds (this condition holds for suffi-ciently long fillings by Lemma 4.1).We then choose the construction of spiderwebs { W i } i ∈ N as in Theorem 4.3, andspecifically the family constructed via Proposition 4.7. By Theorem 4.3 we have,for each i , K = (cid:91) i K W i , and by Proposition 4.7 we know that K W i +1 = K W i ∗ (cid:18) ∗ c ∈ E i K c (cid:19) , for some finite E i ⊂ C , where K c is a conjugate of some filling kernel N j . It followsthat(1) K W i = ∗ c ∈ E i K c , where E i = i (cid:71) j =1 E j . Since K is an increasing union of the subgroups K W i , we have, for T = ∪ i E i = (cid:116) i E i K = ∗ c ∈ T K c . It remains to show that T meets each K –orbit in C exactly once. Since (cid:83) i W i = X , it is clear that each K –orbit of element of C is eventually included in one of the E i . Suppose by contradiction that there is k ∈ K and c , c ∈ T so that kc = c .Let j be chosen large enough so that c , c ∈ E j , and so that k ∈ K W j . Then thesubgroups K c and K c are conjugate inside K W j , contradicting the free productstructure (1). (cid:3) In [DGO, Theorem 7.9] it is proved that the kernel is a free product of conjugatesof the filling kernels N i . The only new part of the above result is to identify theindexing set for the free product as being in bijection with the K –orbits of C . Webelieve that this description of the indexing set also follows from the construction ofwindmills in [DGO], and also that this description is surely known by the authorsof [DGO]. 5. Approximating the boundary of a Dehn filling
The statements in this section form the core of our new method for understandingthe boundary of a Dehn filling. In this section we give statements in the absoluteand relative setting, but only use (or indeed prove) the absolute statements in thesequel. The careful reader will see that the relative statements are strictly easierto establish.The absolute (hyperbolic) statements require some further constructions, whichwe give in the next subsection.
OUNDARIES OF DEHN FILLINGS 21
Truncated quotients.
Let ( G, P ) be relatively hyperbolic. In subsequentsections we will focus on (long) filling kernels { K i (cid:67) P i } with P i /K i hyperbolic foreach i . We call such fillings hyperbolic fillings . Since we do not require anythingabout the hyperbolicity constant of (a Cayley graph of) P i /K i , we do not get auniform hyperbolicity constant for quotients of (a given Cayley graph of) G by thefilling kernel. We overcome this by taking truncated quotients as defined below.Having a uniform hyperbolicity constant regardless of the long hyperbolic fillingwill be crucial for us. Recall that in the case that P i /K i is (virtually) Z , thecorresponding truncated horoball can be thought of as (the lift to the universalcover of) a Margulis tube, as discussed in Subsection 2.4.Let K be the normal closure in G of (cid:83) i K i . For a sufficiently long filling, it isthe case that the intersection K c of K with a horoball stabilizer is conjugate tosome K i (see Theorem 2.14.(1)). If we are assuming that the P i /K i are hyperbolic,this means that K c acts on each “horosphere” H Dc ⊂ H c with quotient a Gromovhyperbolic graph. We saw in Subsection 2.4 that for sufficiently deep horospheres,the quotient is a hyperbolic graph with uniform constant.Fix a cusped space X for ( G, P ). In Section 4, we constructed, for sufficientlylong fillings G → G = G ( K , . . . , K n ) a sequence of spiderwebs (Definition 4.2) W k ⊆ W k +1 ⊆ · · · ⊂ X , each of which is stabilized by a subgroup K W j of thekernel of G → G .Recall that in this section we are assuming all the quotients P i /K i are hyperbolicgroups. The universal constant θ comes from Corollary 2.20. Definition 5.1.
Let W be a θ –spiderweb in X , associated to the filling kernels { K i (cid:67) P i } . The truncated quotient T W associated to W is obtained in the followingway. As in Subsection 2.4, for each horoball center c , we we let t c be the minimalinteger so that the quotient by K c of the horosphere at depth t c in H c satisfiesGromov’s 4–point condition Q (5), and note that Corollary 2.20 then implies thatthe quotient H [500 θ,t c ] c /K c is θ –hyperbolic (where H [500 θ,t c ] c is that part of thehoroball H c between depth 500 θ and t c ). Let Σ c denote the horosphere at depth t c , centered at c . The group K W acts properly on X \ (cid:83) {H ( t c , ∞ ) c | c ∈ C ( W ) } ,and we let T W be the quotient by this action. We similarly define T G to be the thequotient of X \ (cid:83) {H ( t c , ∞ ) c | c ∈ C} by K .The points coming from W in T W form a compact subset. The quotient T G isquasi-isometric to G .5.2. Statements for hyperbolic fillings.
The next results are some of the mainingredients of the proof of Theorem 1.2.The following theorem says that, for W a θ –spiderweb associated to a long filling, T W is hyperbolic and visual (as in Definition 3.4) with uniform constants, and itdescribes the topology of its boundary. Theorem 5.2.
Let ( G, P ) be relatively hyperbolic with cusped space X . Then thereexist θ, δ with the following properties. For all sufficiently long hyperbolic fillings G → G = G ( N , . . . , N n ) with N i infinite for all i and any θ –spiderweb W (seeDefinition 4.2) associated to the filling we have(1) The truncated quotient T W is δ –hyperbolic and δ –visual, and so is T G . (2) If F is the union of subsets of ∂T W of the form Λ(Σ c /K c ) for c ∈ C ( W ) ,then there exists a regular covering map ( ∂X \ Λ( K W )) → ∂T W \ F withdeck group K W .(3) ∂T W \ F is open and dense in ∂T W . The following theorem describes the boundary of the quotient group as a limitof boundaries of T W i , where the θ –spiderwebs W i form an exhaustion of the cuspedspace. Recall the notion of weak Gromov–Hausdorff convergence from Section 3. Theorem 5.3.
Let ( G, P ) be relatively hyperbolic. Then there exist (cid:15), κ so that forall sufficiently long hyperbolic fillings G → G = G ( N , . . . , N n ) with N i infinite forall i the following hold:(1) For any θ –spiderweb W as in Theorem 5.2 associated to the filling there isa visual metric ρ W on ∂T W based at the image of with parameters (cid:15), κ .(2) If W ⊆ W ⊆ . . . is a sequence of θ –spiderwebs with (cid:83) W j = X , then thesequence (cid:8) ( ∂T W j , ρ W j ) (cid:9) weakly Gromov–Hausdorff converges to a visualmetric on ∂T G . The following theorem guarantees that the boundaries of the T W are linearlyconnected with uniform constant. This is important in order to be able to applyLemma 3.9. Theorem 5.4.
Let ( G, P ) be relatively hyperbolic and suppose that the Bowditchboundary ∂X (when endowed with any visual metric) is linearly connected. Thenfor all sufficiently long hyperbolic fillings G → G = G ( N , . . . , N n ) with G one-ended and N i infinite for each i , the following holds: There exists L so that, forany θ –spiderweb W with parameter θ as in Theorem 5.2, ( ∂T W , ρ W ) is L –linearlyconnected, where ρ W is the visual metric from Theorem 5.3. Note that L depends on the filling, but then is uniform over θ –spiderwebs asso-ciated to that filling. The constants δ , (cid:15) , κ do not depend on the (long) filling.5.3. Statements for general fillings.
For a general (not-necessarily-hyperbolic)long Dehn filling, we can make similar statements as above for the Bowditch bound-ary of ( G, P ). We use the following terminology. For W a spiderweb associatedto a Dehn filling, let X W = X/K W . If K is the kernel of the filling map G → G ,let X G = X/K , and note that, for long fillings, X G is a cusped space for the pair( G, P ). In particular, ∂ ( G, P ) = ∂X G . Theorem 5.5.
Let ( G, P ) be relatively hyperbolic with cusped space X . Then thereexist θ, δ with the following properties. For all sufficiently long fillings G → G = G ( N , . . . , N n ) with N i infinite for all i and any θ –spiderweb W associated to thefilling we have(1) The quotient X W is δ –hyperbolic and δ –visual, and so is X G .(2) If F ⊂ ∂X W consists of the points Λ( H c /K c ) for c ∈ C ( W ) , then thereexists a regular covering map ( ∂X \ Λ( K W )) → ∂X W \ F with deck group K W .(3) Let F iso ⊆ F consist of those points which are isolated in ∂X W (so theycome from finite index N i (cid:67) P i ). Then ∂X W \ F is open and dense in ∂X W \ F iso . OUNDARIES OF DEHN FILLINGS 23
Theorem 5.6.
Let ( G, P ) be relatively hyperbolic. Then there exist (cid:15), κ so that forall sufficiently long fillings G → G = G ( N , . . . , N n ) with N i infinite for all i thefollowing hold:(1) For any θ –spiderweb W as in Theorem 5.5 associated to the filling there isa visual metric ρ W on ∂X W based at the image of with parameters (cid:15), κ .(2) If W ⊆ W ⊆ . . . is a sequence of θ –spiderwebs with (cid:83) W j = X , then thesequence (cid:8) ( ∂X W j , ρ W j ) (cid:9) weakly Gromov–Hausdorff converges to a visualmetric on ∂X G . Theorem 5.7.
Let ( G, P ) be relatively hyperbolic and suppose that the Bowditchboundary ∂X (when endowed with any visual metric) is linearly connected. Thenfor all sufficiently long fillings G → G = G ( N , . . . , N n ) with ∂ ( G, P ) linearlyconnected and N i infinite for each i , the following holds: There exists L so that, forany θ –spiderweb W with parameter θ as in Theorem 5.5, ( ∂X W , ρ W ) is L –linearlyconnected, where ρ W is the visual metric from Theorem 5.6. Proofs of approximation theorems for hyperbolic fillings
In this section we give proofs of the hyperbolic versions of the theorems statedin the last section, namely Theorems 5.2, 5.3 and 5.4. No other results from eitherthis section or the last are used in the sequel.
Assumption 6.1. [Choice of θ ] We fix once and for all a choice of θ so that:(1) θ ≥ θ is a hyperbolicity constant for the cusped space X , and X is also θ –visual;and(3) θ ≥ θ , where θ is as in Corollary 2.20.From now on we drop “ θ ” when talking about spiderwebs.6.1. Hyperbolicity and visibility of the truncated quotient.
We now proveTheorem 5.2.(1) which states that the truncated quotient T W is δ –hyperbolic and δ –visual, and so is T G , for some constant δ which is independent of the (long) fillingand the spiderweb W .Both δ –hyperbolicity and δ –visibility are proved using a kind of local-to-globalprinciple. In the case of hyperbolicity, this is the Coarse Cartan–Hadamard Theo-rem of Delzant and Gromov [DG08]. We use the formulation of Coulon in [Cou14].Say that a space is r –simply-connected if the fundamental group is normally gener-ated by free homotopy classes of loops of diameter less than r . Theorem 6.2 (Coarse Cartan–Hadamard) . [Cou14, A.1] Let ν ≥ , and let R ≥ ν . Let M be a geodesic space. If every ball of radius R in M is ν –hyperbolicand if M is − R –simply-connected, then M is ν –hyperbolic. The following is our local-to-global principle for visibility. (Recall from Definition3.4 that a space is visual if, roughly speaking, geodesics can be coarsely prolongedto geodesic rays.)
Proposition 6.3 (Local visibility implies global visibility) . For every ν ≥ thefollowing holds. Let M be a proper ν –hyperbolic space and suppose that for all p, q ∈ M with d ( p, q ) ≤ ν there exists a geodesic [ p, q (cid:48) ] of length at least ν with d ( q, [ p, q (cid:48) ]) ≤ ν . Then M is ν –visual. Proof.
Let [ a, b ] be a geodesic segment. We will define a sequence of points { p i } i ∈ Z ≥ so that (i) p = a and b ∈ { p , p } ; (ii) the Gromov products ( p i − , p i +1 ) p i aresmall: and (iii) the distances d M ( p i +1 , p i ) are large for i >
0. A standard argu-ment (eg [AGM16, Lemma 4.9]) then shows that the concatenations of geodesics[ p , p ] · · · [ p n − , p n ] lie close to any geodesics [ p , p n ]. These geodesics (sub)convergeto a geodesic ray passing near b .Two cases must be distinguished. If d M ( a, b ) < ν , then let p = b . Now choosea geodesic σ of length 200 ν beginning at a and passing within ν of b . Choose p to be the point on σ at distance 50 ν from b .The second case is when d M ( a, b ) ≥ ν . In this case, choose p to be the pointon [ a, b ] at distance 50 ν from b and let p = b .We can inductively suppose that points p , . . . , p i − have been chosen, and that d M ( p i − , p i − ) = 50 ν . We apply the hypothesis of the lemma with p = p i − and q = p i − to find a geodesic σ i of length 200 ν beginning at p i − , passing within ν of p i − . The geodesic σ i contains a point at distance 50 ν from p i − and distance atleast 98 ν from p i − . We choose p i to be such a point of σ i .We thus have a sequence of points p , p , . . . so that d M ( p i , p i +1 ) = 50 ν and d M ( p i , [ p i − , p i +1 ]) < ν for each i >
0. A standard argument shows the concate-nation [ p , p ] · · · [ p n − , p n ] lies in a 5 ν –neighborhood of [ p , p n ]. In particular, thepoint b lies within 5 ν of any geodesic [ p , p n ]. Because M is proper, a sequenceof geodesics γ n = [ p , p n ] must subconverge to a geodesic ray γ which also passeswithin 5 ν of the point b . (cid:3) We want to consider a long hyperbolic filling of ( G, P ), and the truncated partialquotient T W associated to a spiderweb for the kernel of such a filling (see Definition5.1). In particular, we will show it is δ –hyperbolic and δ –visual, where δ = 1500 θ ,thus establishing Theorem 5.2.(1). Claim 6.3.1. T W is 100 θ –simply-connected. Claim 6.3.2.
For all sufficiently long hyperbolic fillings G → G , and any associatedspiderweb W , the following holds. Let B be a ball of radius 10 θ in T W or T G .Then B is isometric to a ball in either X or H [500 θ,t c ] c /K c for some c ∈ C . Moreover,the first case holds whenever B is not entirely contained in a horoball.Before proving Claims 6.3.1 and 6.3.2, we argue that together they imply hyper-bolicity and visibility of T W (the argument for T G is identical). Since both X and H [500 θ,t c ] c /K c are θ –hyperbolic (see Corollary 2.21), it follows immediately from thetwo claims and Theorem 6.2 that T W is 300 θ –hyperbolic.We now check that T W is 1500 θ –visual using Proposition 6.3 with ν = 300 θ (the argument for T G is identical). Consider a geodesic [ p, q ] in T W with d ( p, q ) ≤ ν = 3 · θ . We need to find a geodesic [ p, q (cid:48) ] of length at least 6 · θ andso that d ( q, [ p, q (cid:48) ]) ≤ ν = 300 θ . If p is contained in a ball B in T W of radius 10 θ isometric to a ball in X , then the required geodesic of length 6 · θ exists since X is θ –visual. If not, a ball of radius 10 θ centered at p lies in some horoball H [500 θ,t c ] c /K c . In particular p lies at depth at least 10 θ , and [ p, q ] lies in a horoball H [10 θ,t c ] c /K c . Lemma 2.23 gives a geodesic [ p, q (cid:48) ] passing within 2 θ < ν = 300 θ of q . By Proposition 6.3, T W is 5 ν = 1500 θ –visual. OUNDARIES OF DEHN FILLINGS 25
Proof of Claim 6.3.1.
Note that X is θ –hyperbolic. It follows immediately from[BH99, III.H.2.6] that X is 16 θ –simply-connected, which is to say that π ( X ) isnormally generated by free homotopy classes of loops of length at most 16 θ .Now π ( X W /K W ) is normally generated by free homotopy classes of:(1) The images of the loops which normally generate π ( X ); and(2) Loops representing a choice of generators of K W .We can choose loops which represent generators of K W to each lie within a singlehoroball at depth 0.The subgraph T W of X/K W agrees with X/K W at depth less than t c (for eachgiven horoball), so the generators of π ( T W ) can be taken to be a collection of loopswhich are either:(1) Images of loops representing generators of π ( X ) of length at most 16 θ ; or(2) Peripheral loops, entirely contained in a horoball.Any path in a horoball can be pushed across pentagons and squares to maximaldepth. Since at maximal depth the horoball is θ –hyperbolic, another applicationof [BH99, III.H.2.6] implies that T W is 100 θ –simply-connected, as required. Thisproves Claim 6.3.1. (cid:3) Proof of Claim 6.3.2.
We suppose that G → G is a long enough filling to applyCorollary 2.25 with C = 10 θ and to apply Lemma 2.16 with R = 10 θ . Inparticular, we have t c ≥ θ for all c ∈ C . We fix an associated spiderweb W ,and prove the Claim for T W , the proof for T G being almost identical.Let B (cid:48) be a ball with the same center as B and radius 10 θ . Notice that geodesicsconnecting points in B are contained in B (cid:48) . We distinguish two cases.In the first case, B (cid:48) is disjoint from G/K W , the image of the Cayley graph in X/K W . In this case, B is isometric to a ball in H [500 θ,t c ] c /K c .In the second case, B (cid:48) intersects G/K W , say at the image of p ∈ G . Since t c ≥ θ , B misses the truncation completely, and is isometric to a ball ˆ B in X/K W with center at depth ≤ θ . This ball ˆ B is entirely contained in the imageof a ball B (cid:48) of radius 10 θ centered on a vertex of the Cayley graph of G contained in X . Using Lemma 2.16, this ball embeds isometrically into X/K W , so B is actuallyisometric to a subset of B (cid:48) ⊂ X . This proves Claim 6.3.2. (cid:3) As we explained above, these two claims imply Theorem 5.2.(1), so the proof ofthis theorem is complete.
Assumption 6.4. [Choice of δ ] For the remainder of this section, δ denotes theconstant in Theorem 5.2.(1); i.e. δ = 1500 θ .6.2. Topology of the boundary of the truncated quotient.
In this section,we prove part (2) of Theorem 5.2 about the existence of a covering map ( ∂X \ Λ( K W )) → ∂T W \ F with deck group K W (for appropriate set F ). To this end,we fix for this subsection a hyperbolic Dehn filling of ( G, P ) with associated θ –hyperbolic cusped space X , and make the following assumption. Assumption 6.5.
The Dehn filling is sufficiently long so that:(1) If W is a spiderweb associated to the filling, then the truncated quotient T W is δ –hyperbolic and δ –visual (Theorem 5.2.(1))(2) Every truncation depth t c is at least 10 θ (Corollary 2.25). (3) For any x ∈ X and any k ∈ K \ { } , any geodesic [ x, kx ] meets somehoroball at depth D ≥ θ (Lemma 2.26).We also fix a spiderweb W ⊂ X associated to the Dehn filling we have chosen. Definition 6.6.
The saturated spiderweb S W is W ∪ (cid:32) (cid:83) c ∈C ( W ) (cid:98) H c (cid:33) . The truncated,quotiented version S W is the intersection of S W /K W with T W ⊆ X/W .6.2.1.
Quasiconvexity and limit sets.
Lemma 6.7.
The saturated spiderweb S W is θ –quasiconvex in X .Proof. If A is a K –quasiconvex set, and W is a collection of L –quasiconvex sets,each of which has nonempty intersection with A , an easy quadrangular argumentshows that A ∪ (cid:83) W is (max { K, L } + 2 θ )–quasiconvex.The spiderweb W is 4 θ –quasiconvex, and the horoballs (cid:98) H c are 0–quasiconvex,so the result follows. (cid:3) Since X → X/K W is a covering map and S W is K W –equivariant, we have thefollowing corollary. Lemma 6.8. S W /K W is θ –quasiconvex in X/K W . Next we show that the quasiconvexity persists after we truncate.
Lemma 6.9. S W is δ –quasiconvex in T W .Proof. Suppose that p, q ∈ S W . Let γ be a X/K W –geodesic between p and q .According to [GM08, Lemma 3.10], we may assume that γ intersects any horoballin a path which consists of at most two vertical segments and a single horizontalsegment. We form a path γ in T W between p and q as follows. Any part of γ whichis not contained in T W lies in a truncated part of a horoball H . Such a segmentof γ consists of two vertical segments (of length at least t c ) and a single horizontalsegment. Replace any such subsegment below depth t c by a geodesic at depth t c inthe truncated horoball.Applying Lemma 6.8 to γ and noting that γ \ γ lies entirely in S W , we see that γ lies in a 6 θ –neighborhood of S W .We claim that γ is a 10 δ –local geodesic in T W . At depths less than t c − δ thisis clear, since at such depths the spaces X/K W and T W , and the paths γ and γ ,are locally identical. Thus suppose that σ is a subsegment of γ of length 10 δ thathas at least one point at depth greater than t c − δ . Since t c (cid:29) δ (Assumption6.5.(2) above), this subsegment lies entirely inside a single truncated horoball.Let a and b be the endpoints of σ . If σ were part of the original path γ thensince distances in T W are greater than those in X/K W , σ remains a geodesic inthis case.We are left with the possibility that σ is not part of the original path γ . Supposethat σ is a T W –geodesic between a and b , chosen to satisfy the conclusion ofLemma 2.22. In particular, σ consists of at most two vertical segments and asingle horizontal segment, either at depth t c or having length at most 3. It is clearthat the only way σ could be shorter than σ is if σ does not intersect depth t c , since in every other case the only possible difference between σ and σ is thechoice of geodesic at depth t c . However, if σ does not intersect depth t c , it must OUNDARIES OF DEHN FILLINGS 27 be that neither a nor b is at depth t c , and the X/K W –geodesic between a and b goes beneath depth t c (in order that p be truncated). Lemma 2.22 ensures thatany horizontal segment in σ has depth at most 3. But then it is clear that therewould not have been truncation, since σ is then an X/K W –geodesic as well. Thus σ cannot be shorter than σ and we have argued that γ is a 10 δ –local geodesic, asrequired. By [BH99, III.H.1.13], any geodesic joining p to q lies within 2 δ of such apath, and thus lies in the (2 δ + 6 θ )–neighborhood of S W . Since 6 θ < δ , the lemmais proved. (cid:3) Lemma 6.10. Λ( S W ) = Λ( W ) = Λ( K W ) Proof.
The group K W stabilizes W , hence Λ( K W ) ⊆ Λ( W ), and W ⊆ S W , soΛ( W ) ⊆ Λ( S W ). It remains to show Λ( S W ) ⊆ Λ( K W ). Let { x i } be a sequence ofpoints in S W converging to some x ∈ ∂X . We can assume that either (i) they areall contained in G ; or (ii) each is contained in a horoball (cid:98) H g i c for some g i ∈ K W (there are finitely many K W –orbits of horoballs intersecting W ). In the first case, x ∈ Λ( K W ) because K W acts cocompactly on W ∩ G . In the second case, up topassing to a subsequence one of the following holds: Either all g i c coincide or all g i c are pairwise distinct.First suppose that all g i c coincide. Recalling that c is the point at infinity of (cid:98) H c , we have x = g i c , and g i c ∈ Λ( K W ) because K g i c < K W is infinite.Finally, suppose that all g i c are pairwise distinct. In this case it is easy to seethat x coincides with the limit of the g i , hence x ∈ Λ( K W ) as required. (cid:3) We next describe the limit set of S W . Recall that the set F is the union of thelimit sets Λ(Σ c /K c ) for c ∈ C ( W ). Lemma 6.11. Λ( S W ) is F .Proof. Note that S W is finite Hausdorff distance from the union (cid:83) c ∈C ( W ) Σ c /K c ,which (choosing representatives of K W –orbits of horoball centers c ) is actually afinite union of quasiconvex sets of the form Σ c /K c . The limit set Λ( S ) is thus equalto the union of the limit sets of the Σ c /K c in ∂T W , which is F . (cid:3) The action of K W on S W and X . Definition 6.12.
The frontier of S W is the set of vertices in S W which are joinedby an edge to a point in X \ S W .Observe that by construction every vertex in the frontier of S W has depth atmost 500 θ . Lemma 6.13.
Suppose that x (cid:54)∈ S W or x belongs to the frontier of S W . Then forany k ∈ K \ { } we have d ( x, kx ) > δ .Proof. Consider x (cid:54)∈ S W . Then any y ∈ π S W ( x ) is contained in the frontier of S W . By Assumption 6.5.(3), any geodesic [ y, ky ] must go at least 10 θ into somehoroball. Since the depth of y is at most 500 θ , we have d ( y, ky ) ≥ · (10 − θ .The broken geodesic [ x, y ] ∪ [ y, ky ] ∪ [ ky, kx ] is 2 θ close to a geodesic [ x, kx ], fromwhich it quickly follows that d ( x, kx ) > . · θ = 100 δ , as required. (cid:3) Corollary 6.14.
Let x be a point in X \ N δ ( S W ) . Then the map from X to X/K W restricts to an isometry from the δ –ball in X around x to a δ –ball in T W . The covering map.
Let φ : X → X/K W be the quotient map. We define amap Θ : ( ∂X \ Λ( K W )) → ( ∂T W ) \ F as follows. Represent ξ ∈ ∂X \ Λ( K W ) by a 50 δ –local geodesic ray γ : [0 , ∞ ) → X starting at 1. (Note that there is a geodesic ray Hausdorff distance at most 3 θ from γ ; we use local geodesic rays because they occur naturally in the proof anyway.)Let R γ be the smallest number so that d ( γ ( t ) , S W ) ≥ δ for all t ≥ R γ . Since ξ / ∈ Λ( K W ) = Λ( S W ) (see Lemma 6.10), and S W is quasiconvex, there is such an R γ . If γ is actually a geodesic, Lemma A.12 can be used to show that γ makeslinear progress away from S W after time R γ :(2) 100 δ − θ + t < d ( γ ( R γ + t ) , S W ) ≤ δ + t. Similar statements can be made for a 50 δ –local geodesic, using the fact it is quasi-geodesic and close to a geodesic, and/or replacing 100 δ with any quantity sufficientlylarge with respect to θ .Define γ ( t ) = φ ( γ ( t + R γ )). The image γ of the ray in X/K W lies entirely in T W \ N δ (cid:0) S W (cid:1) . It follows from Corollary 6.14 that γ is a 50 δ –local geodesic inthe δ –hyperbolic space T W . In particular, [ γ ] represents a point of ( ∂T W ) \ F , andwe define Θ( ξ ) to be this point. Lemma 6.15.
The map Θ is well-defined and continuous.Proof. Suppose γ and { γ i } i ∈ N are 50 δ –local geodesics in X starting at 1, so that[ γ i ] → [ γ ] in ∂X . We show that Θ is well-defined and continuous by showing that[ γ i ] → [ γ ] in ∂T W . (To deduce the map is well-defined, take a constant sequence.)If all the rays are geodesic, they stay within θ of each other on larger and largerinitial subsegments. Using the inequality (2) above, it can be shown that for allbut finitely many i , we have d ( γ i ( R γ i ) , γ ( R γ )) ≤ θ . If they are only 50 δ –localgeodesics, we can use the fact that they are 3 θ –close to geodesics to get the bound d ( γ i ( R γ i ) , γ ( R γ )) ≤ θ < δ for all but finitely many i . It follows that the rays γ and { γ i } all start within δ ofone another. Moreover, for any large t , all but finitely many γ i pass within 7 θ of γ ( t ). It follows that the equivalence classes { [ γ i ] } converge to [ γ ], as required. (cid:3) Proposition 6.16.
The map Θ is a covering map. The preimage of any givenpoint is a K W –orbit in ∂X .Proof. Let ξ ∈ ( ∂T W ) \ F , and let γ be a geodesic ray in T W from 1 to ξ . For t sufficiently large there exists an open neighborhood U ⊆ ( ∂T W ) \ F of ξ with theproperty that any 50 δ –local geodesic from N δ ( S W ) to U passes within 10 δ of b = γ ( t ).Let { b g } g ∈ K W be the preimage of b , with indexing so that gb h = b gh for all g, h ∈ K W . Let R g be the set of all lifts to X starting at b g of 50 δ –local geodesic raysin T W starting at b and limiting to U (notice that T W is a subset of X/K W whichis covered by X ). Since S W is quasiconvex, and b starts away from N δ ( S W ),we can apply Corollary 6.14 to imply that all such lifts are 50 δ –local geodesic rays.Let U g be the set of all limit points in ∂X of elements of R g .Clearly, for any g, h ∈ K W we have gU h = U gh . We now have to prove that(1) for each distinct g, h ∈ K W we have U g ∩ U h = ∅ , OUNDARIES OF DEHN FILLINGS 29 (2) for each g ∈ K W , U g is open and Θ | U g is a homeomorphism onto U ,(3) Θ − ( U ) = (cid:83) g ∈ K W U g .To show item (1), notice that any γ ∈ R g has the property that the diameterof π S W ( γ ) is at most 20 δ . For g, h distinct elements of K W , the distance between π S W ( b g ) and π S W ( b h ) = hg − π S W ( b g ) is at least 100 δ by Lemma 6.13. In particular,elements of R g cannot be asymptotic to elements of R h , so U g ∩ U h = ∅ .Let us prove that U g is open. Let [ α ] ∈ U g . We may suppose α ∈ R g , so α is thelift of α starting at b g and α is a 50 δ –local geodesic starting at b and limiting to apoint in U . Since U is open, some standard basic neighborhood of [ α ] is contained in U . In particular, for t chosen sufficiently large, if η is a 50 δ –local geodesic startingat b and passing within 10 δ of α ( t ), then [ η ] ∈ U .Now consider a standard neigborhood V of [ α ] ∈ ∂X , the set of points repre-sented by 50 δ –local geodesic rays starting at b g and passing within 10 δ of α ( t ). Bythe previous paragraph, all such rays are elements of R g , so V ⊆ U g . Since [ α ] wasarbitrary, this proves U g is open.Let us prove that Θ | U g is injective. Take γ , γ ∈ R g with distinct limit pointsin ∂X . Then for some smallest t , d ( γ ( t ) , γ ( t )) ≥ δ , which implies that thesame holds for their projections to T W (see Corollary 6.14), which in turn impliesthat such projections have distinct limit points.Surjectivity of Θ | U g onto U and continuity of the inverse are clear from thedefinition via lifts. We proved item (2).We are left to prove that for any ray α in X starting at 1 with Θ( α ) ∈ U wehave α ∈ U g for some g . Let α (cid:48) be the subray of α that intersects N δ ( S W )at its starting point only. By the defining property of b , the projection α (cid:48) of α (cid:48) to T W passes within 10 δ of b , which implies that α (cid:48) passes within 10 δ of b g forsome g ∈ K W . Any geodesic ray starting at b g and asymptotic to α belongs to R g because its projection to T W is asymptotic to α (cid:48) which limits to U . Hence, the limitpoint of α is in U g , as required. This completes the proof of Proposition 6.16. (cid:3) Proof of Theorem 5.2. (2) . We defined a map Θ : ( ∂X ) \ Λ( K W ) → ( ∂T W ) \ F inSubsection 6.2.3 and proved that it is a covering map in Proposition 6.16. The factthat this covering is regular with deck group K W may be seen as follows: For any k ∈ K W and ξ ∈ ( ∂X \ Λ( K W )) we have Θ( kξ ) = Θ( ξ ) since we can represent ξ and kξ by rays γ and kγ , so that the definition of Θ and Lemma 6.15 clearly giveΘ( ξ ) = Θ( γ ) = Θ( kγ ) = Θ( kξ ). Hence, K W acts by deck transformations, and bythe description of preimages of points given by Proposition 6.16, it acts transitivelyon preimages of points. Hence, the covering is regular with deck group exactly K W . (cid:3) Connectedness of ∂T W . The following result is Theorem 5.2.(3).
Lemma 6.17.
For any spiderweb W associated to a sufficiently long filling, ∂T W \F is open and dense in ∂T W .Proof. We have already remarked that F is a finite union of closed sets, so ∂T W \ F is open.More specifically, the set F is a finite disjoint union of closed sets F , . . . , F k ,where each F i is the limit set of some Σ c /K c . Let c , . . . , c k be representatives of the K W –orbits of the points c which occur, and write F i = Λ(Σ c i /K c i ), so F = (cid:116)F i . Let ξ ∈ F i be represented by a geodesic ray γ . By quasiconvexity, we mayassume that some tail of γ is contained in the truncated image of (cid:98) H c i . Moreover,we may assume this tail is entirely horizontal. For N ∈ Z very large, we form anew, 50 δ –local geodesic ˆ γ N which agrees with γ up to t = N , and then changes toa vertical path until it leaves the image of (cid:98) H c . Using δ –visibility, this path is closeto a geodesic ray γ N which fellow travels γ for time N , but tends to a point not inthe limit set of F i . Some of these γ N may end up in F j for j (cid:54) = i , but each F j isdisjoint from some open neighborhood of F i , so this only happens for finitely many N . It follows that ξ is a limit of points in ∂T W \ F . (cid:3) We have now proved Theorem 5.2.
Corollary 6.18.
Suppose W is associated to a sufficiently long filling and that ∂X \ Λ( K W ) is connected. Then ∂T W is connected.Proof. Theorem 5.2.(2) implies that ∂T W \ F is covered by ∂X \ Λ( K W ). Thusconnectedness of ∂T W \ F follows from connectedness of ∂X \ Λ( K W ). By Lemma6.17, ∂T W \ F is open and dense in ∂T W . It follows that ∂T W is connected. (cid:3) Choosing the visual metric.
We have already proved (Theorem 5.2.(1))that, for long fillings, the T W are all δ –hyperbolic, for a fixed δ >
0. Fix (cid:15) = δ .Fix κ = κ ( (cid:15), δ ) as in Proposition 2.3.Fix a spiderweb W with parameter θ associated to a filling sufficiently long that T W is δ –hyperbolic and δ –visual. We denote the image in T W of 1 ∈ X by 1.Proposition 2.3 implies that there exists a visual metric ρ W ( · , · ) on ∂T W based at1 with parameters (cid:15), κ . This proves Theorem 5.3.(1).6.4. Convergence.
In this subsection, we prove Theorem 5.3.(2), which statesthat the visual metrics constructed in the last subsection weakly Gromov–Hausdorffconverge to a visual metric on ∂T G . Proof of Theorem 5.3. (2) . Using Proposition 3.5, it is enough to show that the T W j strongly converge to T G , a space on which G acts geometrically. Fix any R ≥ j be large enough that W j contains the ball B of radius 2 R in X , and moreoverwhenever x, y ∈ B are in the same K –orbit then they are in the same K W j –orbit.The latter property can be arranged since there are only finitely many k ∈ K sothat there exists x ∈ B with kx ∈ B .We will show that there exists a locally isometric bijection b that preserveslengths of paths from the ball B j of radius 2 R around 1 in T W j to the ball B of radius2 R around 1 in T G . Such bijection restricts to an isometry on the correspondingballs of radius R , proving strong convergence since R was arbitrary.Before defining b , notice that there are covering maps Φ j : X \ (cid:83) {H ( t c , ∞ ) c | c ∈C ( W ) } → T W j and Φ : X \ (cid:83) {H ( t c , ∞ ) c | c ∈ C} → T G (the domains of the twocovering maps are obtained from X by removing different sets of horoballs).The map b is defined by b ( x ) = Φ(Φ − j ( x )). In order to show that it is welldefined we have to show that Φ − j ( x ) is contained in the domain of Φ and that anypoint in Φ − j ( x ) has the same image under Φ. In order to show the former propertynotice that, since we can lift geodesics from T W j to X \ (cid:83) {H ( t c , ∞ ) c | c ∈ C ( W ) } topaths of the same length, B j is contained in Φ j ( B ), so that Φ − j ( x ) ⊆ K W j B ⊆ W .Hence, if by contradiction we had some y ∈ Φ − j ( x ) ∩ H ( t c , ∞ ) for some c ∈ C then OUNDARIES OF DEHN FILLINGS 31 we would actually have c ∈ C ( W ), but clearly Φ − j ( x ) ∩ H ( t c , ∞ ) in that case. Thelatter property just follows from the fact that if two points of X are in the same K W –orbit then they are in the same K –orbit.From the fact that b is well-defined and the fact that Φ and Φ j are coveringmaps it follows that b is a local isometry. Injectivity of b follows from the fact thatif two points p, q of K W j B are in the same K –orbit (i.e. Φ( x ) = Φ( y )) then theyare in the same K W j –orbit (i.e. Φ j ( x ) = Φ j ( y )). Surjectivity of b follows from thefollowing argument. If y lies in B , then we can lift a geodesic from 1 to y to a path˜ γ in X \ (cid:83) {H ( t c , ∞ ) c | c ∈ C} of length at most 2 R starting at 1. If x is the endpointof Φ j ◦ ˜ γ , then it is readily checked that b ( x ) = y . The proof that b is a locallyisometric bijection is complete. (cid:3) We have now completed the proof of Theorem 5.3.6.5.
Linear connectedness.
In this subsection we prove Theorem 5.4 about uni-form linear connectedness.
Theorem 5.4.
Let ( G, P ) be relatively hyperbolic and suppose that the Bowditchboundary ∂X (when endowed with any visual metric) is linearly connected. Thenfor all sufficiently long hyperbolic fillings G → G = G ( N , . . . , N n ) with G one-ended and N i infinite for each i , the following holds: There exists L so that, forany θ –spiderweb W with parameter θ as in Theorem 5.2, ( ∂T W , ρ W ) is L –linearlyconnected, where ρ W is the visual metric from Theorem 5.3. We must show that our approximating spaces T W have Gromov boundaries whichare uniformly linearly connected. We first reformulate the linear connectednesscondition in terms of joining points by “discrete paths” of points which are atleast a bit closer. The following is similar to the last part of the proof of [BK05,Proposition 4]. Lemma 6.19.
Let M be a compact metric space. Suppose that there exists L ≥ so that each p, q ∈ M can be joined by a chain of points p = p , . . . , p n = q so that diam( { p . . . , p n } ) ≤ Ld ( p, q ) and d ( p i , p i +1 ) ≤ d ( p, q ) / . Then M is L –linearlyconnected.Proof. Let p, q ∈ M . We can construct a chain of points interpolating between p, q , and then a “finer” one by interpolating between consecutive points of the firstchain, and so on. Formally, we can construct by induction on i sequences of points Q i = { q ij } j =0 ,...,n ( i ) with • Q = { p, q } , • Q i ⊆ Q i +1 , • q i = p, q in ( i ) = q • d ( q i +1 j , Q i ) ≤ Ld ( p, q ) / i , • d ( q ij , q ij +1 ) ≤ d ( p, q ) / i .Define Q to be the closure of (cid:83) Q i , and notice p, q ∈ Q . Also, it is easilyseen that for each q ij we have d ( { p, q } , q ij ) ≤ (cid:80) i − m =0 ( Ld ( p, q ) / m ) ≤ Ld ( p, q ), sothat diam ( Q ) ≤ Ld ( p, q ). Finally, Q is connected because if not one could write Q as a union of disjoint non-empty clopen sets A, B . By compactness we have d ( A, B ) = (cid:15) >
0. Also, since (cid:83) Q i is dense in Q , both A and B intersect Q i for eachsufficiently large i . However, for sufficiently large i and for any q ij , q ij there exists a chain of points in Q i ⊆ Q connecting q ij , q ij where consecutive points are withindistance (cid:15)/ Q = A (cid:116) B . (cid:3) In the specific case that M = ∂Z for a Gromov hyperbolic space Z , and ∂Z is equipped with a visual metric, we can translate this criterion into one aboutgeodesic rays. Since there are many constants involved, we briefly explain theirroles. First of all, δ, κ, (cid:15) are just the usual constants associated to a hyperbolicspace. Secondly, λ needs to be large enough to ensure that, in a sequence of raysinterpolating between two given ones γ , γ , the distance between the limit pointsof consecutive rays is at most one half of the distance between the limit pointsof γ , γ . Finally, the constant S will be the one determining the eventual linearconnectedness constant, which is L . γ α α α n − γ Figure 1.
The criterion of Lemma 6.19 translates into a statement(Lemma 6.20) about rays with certain Gromov products. LargeGromov product corresponds to small distance in the boundary.
Lemma 6.20.
Let δ, κ, (cid:15) > and let λ > ln(2 κ ) /(cid:15) + 10 δ . For every S there exists L with the following property. Let Z be δ –hyperbolic, with a basepoint w and avisual metric ρ on ∂Z based at w with parameters (cid:15), κ . Also, suppose that for eachpair of rays γ , γ starting at w there exists a chain γ = α , . . . , α n = γ of raysstarting at w with ( α i | α i +1 ) w ≥ ( γ | γ ) w + λ and ( α i | γ ) w ≥ ( γ | γ ) w − S . Then ( ∂Z, ρ ) is L –linearly connected.Proof. Notice that λ satisfies(3) κ e − (cid:15)λ +10 (cid:15)δ < . Now fix S , and let L = 10 κ e (cid:15)S +10 (cid:15)δ .We check the criterion in Lemma 6.19. Let Z be δ –hyperbolic, let w be abasepoint, and let ρ ( · , · ) be a visual metric as in the statement of the lemma. Fix OUNDARIES OF DEHN FILLINGS 33 p , q in ∂Z , which we represent by rays γ p , γ q respectively. Let { α i } i =1 ...n be achain of rays with α = γ p , α n = γ q , and satisfying ( α i | α i +1 ) w ≥ ( γ | γ ) w + λ and( α i | γ ) w ≥ ( γ | γ ) w − S . Let p i ∈ ∂Z be the equivalence class of α i .We observed in Section 2 that Gromov products at infinity can be computed,up to a small error, using representative rays. In particular, since ( α i | α i +1 ) w ≥ ( γ p | γ q ) w + λ , we have ( p i | p i +1 ) w ≥ ( p | q ) w + λ − δ , so(4) ρ ( p i , p i +1 ) ≤ κe − (cid:15) ( p | q ) w − (cid:15)λ +10 (cid:15)δ ≤ κ e − (cid:15)λ +10 (cid:15)δ ρ ( p, q ) < ρ ( p, q ) . Similarly, for any p i , we have(5) ρ ( p i , p ) ≤ κ e (cid:15)S +10 (cid:15)δ ρ ( p, q ) = L ρ ( p, q ) . Thus the diameter of the set { p , . . . p n } is at most L ρ ( p, q ). Since p , q werearbitrary, Lemma 6.19 implies that ( ∂Z, ρ ) is L –linearly connected. (cid:3) The following lemma provides a converse to Lemma 6.20 by allowing us to con-struct a sequence of rays starting from an arc in the boundary.
Lemma 6.21.
Let Z be hyperbolic and suppose that ∂Z , when endowed with avisual metric based at w ∈ Z , is linearly connected. Then there exists R > sothat for every C > and every pair of rays γ , γ in Z starting at w there exists asequence of rays γ = α , . . . , α n = γ starting at w with ( α i | α i +1 ) w ≥ ( γ | γ ) w + C and ( α i | γ ) w ≥ ( γ | γ ) w − R .Proof. Denote by δ a hyperbolicity constant for Z and fix w ∈ Z . Then thereexist (cid:15), κ, L and a visual metric ρ based at w with parameters (cid:15), κ so that ( ∂Z, ρ )is L/ R = log( κ L ) /(cid:15) + 20 δ . Fix now any C, γ , γ as inthe statement. Denoting p , p ∈ ∂Z the limit points of γ , γ , there exists an arc I connecting p to p and with diameter ≤ Lρ ( p , p ). Let α be a ray from w toa point p ∈ I . Approximating Gromov products of points at infinity by Gromovproducts of rays, we have e − (cid:15) ( γ | α ) w ≤ κe (cid:15)δ ρ ( p , p ) ≤ κe (cid:15)δ Lρ ( p , p ) ≤ κ e (cid:15)δ Le − (cid:15) ( γ | γ ) w , from which we deduce ( γ | α ) w ≥ ( γ | γ ) w − log( κ L ) /(cid:15) − δ = ( γ | γ ) w − R . Asimilar computation shows that whenever p, q ∈ ∂Z are close enough, any rays γ p , γ q from w to p, q satisfy ( γ p | γ q ) w ≥ ( γ | γ ) w + C . Hence, by a simple compactnessargument, we can find a sequence of points p = a , . . . , a n = p contained in I sothat, for any choice of rays α i from w to a i , { α i } provides the required sequence ofrays. (cid:3) In the current work, we only need the following proposition for a particularvalue of R . However we believe the more general form given will be useful in futurework. Recall that S W denotes the truncated quotient of the saturated spiderweb by K W (see Definition 6.6), while T G denotes the quotient of the cusped space minuscertain horoballs by K (see Definition 5.1). Roughly speaking, we show that a largeneighborhood of S W in T W isometrically embeds in T G . This is a stronger versionof the strong convergence property we used in 5.3.(2). Proposition 6.22.
Let ( G, P ) be relatively hyperbolic, and let X be the corre-sponding θ –hyperbolic cusped space. Then for every R the following holds. Forall sufficiently long hyperbolic fillings G → G and every θ –spiderweb W associ-ated to the filling, N R ( S W ) isometrically embeds into T G . More precisely: Let Φ W : X \ (cid:83) {H ( t c , ∞ ) c | c ∈ C ( W ) } → T W and Φ : X \ (cid:83) {H ( t c , ∞ ) c | c ∈ C} → T G be thenatural covering maps. Then there exists an isometric embedding ι : N R ( S W ) → T G so that ι ◦ Φ W = Φ where both sides are defined (in particular, ι (1) = 1 ). Moreover,the image of ι is N R (Φ( S W \ (cid:83) {H ( t c , ∞ ) c | c ∈ C} )) .Proof. Our proof rests on the following claim.
Claim.
For every R and every sufficiently long filling the following holds. Forevery spiderweb W , whenever g ∈ K and x ∈ X are so that x, gx ∈ X lie in the R –neighborhood of S W , we have g ∈ K W .Let us assume the claim and fix some R ≥ δ . We set R = R + 22 δ and assumethat we are considering a filling sufficiently long that the conclusion of the claimholds and so that t c ≥ R + 500 θ + 1 for every c ∈ C (see Corollary 2.25).Let b : N R ( S W ) → T G be the map defined by b ( x ) = Φ(Φ − W ( x )). First of all, letus check that Φ − W ( x ) is contained in the domain of Φ, and that Φ(Φ − W ( x )) consistsof a single point, so that b is well-defined. The first property follows from the factthat we can lift any geodesic from x to S W to a path of the same length in X ,showing that Φ − W ( x ) is contained in N R ( S W ), where the neighborhood is taken in X . If by contradiction we had y ∈ Φ − W ( x ) ∩ H ( t c , ∞ ) c for some c we would then have H [500 θ, ∞ ) c ∩ S W (cid:54) = ∅ , since t c ≥ R + 500 θ + 1, and hence c ∈ C ( W ). But then clearlyΦ − W ( x ) ∩ H ( t c , ∞ ) c = ∅ , a contradiction. The fact that Φ(Φ − W ( x )) consists of a singlepoint just follows from the fact that if two points are in the same K W –orbit thenthey are in the same K –orbit.We will now show that b is a locally isometric bijection onto its image. The factthat it is locally isometric easily follows from the fact that Φ W and Φ are coveringmaps (and the fact that it is well-defined). Injectivity follows from the Claim, andthe fact that Φ − W ( x ) is contained in N R ( S W ) for each x ∈ N R ( S W ), as we arguedabove.What is more, we claim that for any ball B = B δ ( x ) in T W centered at some x ∈ N R +2 δ ( S W ), b | B is a surjection onto the ball B δ ( b ( x )). In particular, b restrictsto an isometry between balls of radius 10 δ with the same centers. The reason forsurjectivity is simply that we can define an inverse by lifting to X geodesics from b ( x ) to other points in B δ ( b ( x )) and push them to T W using Φ W , obtaining pathsof length at most 20 δ which therefore have endpoints in B .Let ˆ S = b ( N R ( S W )). From what we proved so far, it follows that any pairof points in ˆ S W is connected by a 10 δ –local geodesic contained in b ( N R +2 δ ( S W )).Since any 10 δ –local geodesic stays within 2 δ of any geodesic with the same endpoints(see [BH99, III.H.1.13]), we get that ˆ S W is 4 δ –quasiconvex. (We implicitly used b ( N R +2 δ ( S W )) ⊆ N δ ( ˆ S W ), which follows from the fact that b is 1–Lipschitz sinceit is locally isometric.)Let us now that prove that ι = b | N R ( S W ) : N R ( S W ) → ˆ S W is an isometry. Sinceit is 1–Lipschitz, we are left to show that d ( x, y ) ≤ d ( b ( x ) , b ( y )) for each x, y ∈ N R ( S W ). This holds because any geodesic γ from b ( x ) to b ( y ) is contained in N δ ( ˆ S W ), which in turn is contained in b ( N R ( S W )) (this follows from the statementabout 10 δ –balls above). In particular, x and y are connected by a path of lengthat most d ( x, y ), namely b − ( γ ), as required.Finally, to prove the “moreover” part one just needs to once again consider liftsof geodesics to Φ( S W \ (cid:83) {H ( t c , ∞ ) c | c ∈ C} ). OUNDARIES OF DEHN FILLINGS 35
We now prove the claim.
Proof of Claim.
Choose a filling sufficiently long that Lemma 2.26 applies with D = R + 10 θ . We argue by contradiction, assuming that x, gx provide a coun-terexample. Since the K –orbit of x is discrete, there exists g (cid:48) ∈ K W g so that d ( x, g (cid:48) x ) is minimal (notice that we still have g (cid:48) x ∈ N R ( S W )). Also, g (cid:48) (cid:54) = 1 be-cause we are assuming g / ∈ K W . By Lemma 2.26, any geodesic [ x, g (cid:48) x ] intersectssome horosphere H Dc . Since S W is 6 θ –quasiconvex (Lemma 6.7), such geodesic iscontained in N R +10 θ ( S W ), implying that S W intersects H θc . In turn, this im-plies that we have K c < K W . But then, for k ∈ K c as in Lemma 2.26, we have d ( x, kg (cid:48) x ) < d ( x, g (cid:48) x ), contradicting the minimality of d ( x, g (cid:48) x ). (cid:3) Having proved the claim, the proof of Proposition 6.22 is complete. (cid:3)
The following elementary lemma is useful in the proof of Theorem 5.4. Thenotation A ∼ C B for quantities A and B indicates A ∈ [ B − C, B + C ]. Lemma 6.23.
Let p , p be points in a δ –hyperbolic space. For i ∈ { , } , let α i , β i be geodesic rays based at p i , so that α is asymptotic to α and β is asymptoticto β . Suppose further that the Gromov products ( p | β ( t )) p and ( p | α ( t )) p arebounded above by a constant C for every large enough t . Then ( α | β ) p ∼ C +8 δ ( α | β ) p + d ( p , p ) .Proof. See Figure 2. Choose points a i ∈ α i and b i ∈ β i far away from p and p so p p α α β β Figure 2.
Estimating the Gromov product at p in terms of theone at p .that d ( a , a ) ≤ δ , d ( b , b ) ≤ δ , and so that ( a i | b i ) p i ∼ δ ( α i | β i ) p i .Now notice that d ( p , α ) and d ( p , β ) are at most C + 2 δ . It follows that d ( a , p ) ∼ C +6 δ d ( a , p ) + d ( p , p ), and similarly d ( b , p ) ∼ C +6 δ d ( b , p ) + d ( p , p ). Combining this with the fact that d ( a , b ) ∼ δ d ( a , b ), we get thedesired estimate. (cid:3) Recall that Theorem 5.4 says that, for sufficiently long one-ended hyperbolicfillings and any spiderweb W associated to such filling, the ∂T W have visual metrics ρ W (of uniform parameters) which are uniformly linearly connected. We only expectuniformity over spiderwebs associated to a fixed filling, not uniformity over fillings. Proof of Theorem 5.4.
The idea here will be to build “discrete paths” joining anytwo points at infinity. This means building, between any two rays to infinity, asequence of interpolating rays satisfying the hypothesis of Lemma 6.20. Given apair of rays in T W , there will be two cases, depending on whether the rays begin to diverge far from S W or not. In the first case, we will exploit the linear connectednessof ∂X ; in the second the linear connectedness of ∂G .We must fix some constants before choosing a filling. As before δ = 1500 θ , (cid:15) = δ and κ are the constants (which depend only on δ ) from Theorem 5.3. Fix λ > ln(2 κ ) /(cid:15) + 10 δ as in Lemma 6.20.By hypothesis ∂X is linearly connected. Recall that Lemma 6.21 provides, fora hyperbolic space Z and a basepoint w ∈ Z , a constant R which governs thebehavior of “discrete paths” of geodesic rays based at w , interpolating between twogiven rays. Claim.
There is a number R X so that the conclusion of Lemma 6.21 applies with R = R X and w any vertex of X at depth less than 201 δ . Proof.
There are finitely many G –orbits of vertices in X of bounded depth. (cid:3) We fix such an R X .Now fix a filling G → G so that all the following hold, for every spiderweb W associated to the filling:(1) The truncated quotient T W is δ –hyperbolic and δ –visual (Theorem 5.2).(2) The boundary ∂T W carries a visual metric ρ W based at 1 with parameters (cid:15), κ (Theorem 5.3).(3) The neighborhood N R X + λ +10 δ ( S W ) isometrically embeds in T G (Proposi-tion 6.22).(4) The Assumptions 6.5 hold. In particular Lemma 6.13 and Corollary 6.14hold.By assumption G is one-ended, so ∂T G ∼ = ∂G is linearly connected [BK05, Proposi-tion 4]. We let R G be the constant R from Lemma 6.21 applied to a visual metricon ∂T G based at 1.Finally we fix a spiderweb W associated to this filling. Recall that we havea natural covering map Φ W : X \ (cid:83) {H ( t c , ∞ ) c | c ∈ C ( W ) } → T W . The followinglemma will allow us to move back and forth more easily between geodesic rays in T W and geodesic rays in X . Lemma 6.24.
Let γ be a path in X which avoids the δ –neighborhood of S W ,and let ¯ γ = Φ W ◦ γ be the projection to T W . Then γ is geodesic if and only if ¯ γ isgeodesic.Proof. Our argument is based on the following claim.
Claim.
If ¯ σ is a T W –geodesic lying outside the 100 δ –neighborhood of S W , thenany lift σ of ¯ σ to X is a geodesic. Proof of Claim.
By Corollary 6.14, σ is a 50 δ –local geodesic. Let σ (cid:48) be a geo-desic with the same endpoints. The space X is θ –hyperbolic, so σ (cid:48) lies in a 2 θ –neighborhood of σ , by [BH99, III.H.1.13]. In particular, σ (cid:48) lies in the domain ofΦ W . If σ were not geodesic, σ (cid:48) would have strictly smaller length, and would projectto a path ¯ σ (cid:48) with the same endpoints as ¯ σ , contradicting the assumption that ¯ σ was geodesic. (cid:3) Now let γ be a path in X avoiding the 102 δ –neighborhood of S W , and let ¯ γ bethe projection of γ to T W . It follows that ¯ γ avoids the 102 δ –neighborhood of S W . OUNDARIES OF DEHN FILLINGS 37
One direction of the Lemma is immediate from the Claim; if ¯ γ is geodesic, thenso is γ .In the other direction, suppose that γ is geodesic. Since the points of γ lie outside N δ S W , we can apply Corollary 6.14 to deduce that ¯ γ is a 50 δ –local geodesic in T W . The endpoints of ¯ γ are therefore joined by a geodesic ¯ σ which lies in a 2 δ –neighborhood of ¯ γ , again using [BH99, III.H.1.13]. Thus ¯ σ lies outside N δ S W .Let σ be a lift of ¯ σ with the same initial point as γ . The Claim implies that σ is ageodesic.We now claim that σ has the same terminal point as γ . Indeed, let p be theterminal point of γ and let q be the terminal point of σ , and suppose p (cid:54) = q . Since p and q project to the same point in T W , there must be some k ∈ K W \ { } sothat q = kp . Let p (cid:48) ∈ π S W ( p ), and let q (cid:48) = kp (cid:48) ∈ π S W ( q ). Lemma 6.13 impliesthat d ( p (cid:48) , q (cid:48) ) > δ . Let η be a geodesic joining p to q . Then η ⊂ N δ ( γ ∪ σ ) liesoutside the 99 δ –neighborhood of S W . The set S W is 6 θ –quasiconvex by Lemma6.7, so we can apply Lemma A.12 to deduce that the diameter of π S W ( η ) is at most9 δ , contradicting d ( p (cid:48) , q (cid:48) ) > δ .Since σ and γ are geodesics with the same endpoints, they have the same length.It follows that ¯ γ has the same length as the geodesic ¯ σ , and is therefore geodesic in T W . (cid:3) We now begin the main argument, which is a verification of the hypothesis ofLemma 6.20 for the space T W with S = max { R X + 100 δ, R G + 10 δ } . Accordingly,we fix γ , γ a pair of rays based at 1 ∈ T W , and look for a sequence of interpolatingrays α i as in Lemma 6.20. Let t = ( γ | γ ) . Case 1. d ( γ ( t ) , S W ) ≥ R X + 10 δ .Let t = sup { t | d ( γ ( t ) , S W ) ≤ δ } , and let x = γ ( t ). We note that thedepth of x is bounded by 500 θ + 200 δ < δ .We let γ (cid:48) be the restriction of γ to [ t , ∞ ). Let D = t − t , and note that D ≥ R X + (10 − δ . Let γ (cid:48) be a broken geodesic following γ (cid:48) for distance D ,and then following a geodesic ray asymptotic to γ . Let T be the tripod γ (cid:48) ∪ γ (cid:48) ,and note that Claim 6.24.1.
All points of T are distance at least 200 δ from S W . Proof of Claim.
This is because otherwise there would be points x , x , x on γ ,appearing in the given order, so that x lies at distance at most 201 δ from S W (just pick x within δ of x ), x lies at distance at least R X + 10 δ from S W (pick x within 10 δ of γ ( t )) and x lies at distance at most 201 δ from S W (pick x δ –close to a point on γ (cid:48) − { x } contained in the 200 δ –neighborhood of S W ). Theexistence of such a triple is easily seen to contradict the fact that N δ ( S W ) is2 δ –quasiconvex, since S W is 3 δ –quasiconvex (Lemma 6.9). (cid:3) Since T is simply connected and Φ W : X \ (cid:83) {H ( t c , ∞ ) c | c ∈ C ( W ) } → T W is acovering map, we can lift T to a tripod T ⊂ X . By Lemma 6.24, the legs of thistripod are geodesic. Let γ be the lift of γ (cid:48) , and let γ be a geodesic ray startingat the same point x , asymptotic to the lift of γ (cid:48) .We claim that the Gromov product ( γ | γ ) x is within 10 δ of D . Indeed, thisGromov product can be estimated to within 2 δ using points on the tripod T . The tripod T (respectively its image T ) is δ –quasiconvex, and lies outside a 200 δ –neighborhood of S W (respectively S W ), so Lemma 6.24 implies the projectionis isometric on T . It’s not hard to see that for s , t sufficiently large, we have( γ ( t ) | γ ( s )) x ∼ δ ( γ | γ ) − t = D .The depth of x was at most 201 δ , and so the depth of x is at most 201 δ . Itfollows that there is a discrete path { α , . . . , α n } of rays based at x interpolatingbetween γ and γ , and satisfying:(1) α = γ and α n = γ ;(2) ( α i | α i +1 ) x ≥ ( γ | γ ) x + λ + 100 δ for all i ; and(3) ( γ | α i ) x ≥ ( γ | γ ) x − R X > δ for all i .A similar argument to the one that proves that any point on T is at distance atleast 200 δ from S W proves the following claim. Claim 6.24.2. No α i meets a 102 δ –neighborhood of S W . Proof of Claim.
First of all, it follows from 3 δ –quasiconvexity of S W that for each t ≥ t we have d ( γ ( t ) , S W ) ≥ t − t + 198 δ . In fact, this is easily deduced from thefact that x lies within δ of any geodesic from γ ( t ) to S W .Notice that if p ∈ X − S W then d ( p, S W ) = d (Φ W ( p ) , S W ), because we canproject to T W a shortest geodesic from p to S W and, vice versa, lift a shortestgeodesic from Φ W ( p ) to S W . In particular, for each t ≥ t we have d ( γ ( t ) , S W ) ≥ t − t + 198 δ .In order to prove that α i does not intersect the 102 δ –neighborhood of S W , wecan now proceed similarly to Claim 6.24.1 and argue that if that was not the casewe could find 3 points along α i so that the middle one is far away from S W but theother ones are close, contradicting quasiconvexity of S W . (cid:3) It follows (using Lemma 6.24 again) that the α i project to geodesic rays α (cid:48) i start-ing at x . We may prepend each such ray with the initial segment of γ terminatingat x , to obtain a broken geodesic α (cid:48)(cid:48) i with Gromov product at x bounded above by δ . Let α = γ , and α n = γ . For i / ∈ { , n } , let α i be a geodesic ray beginning at1 and asymptotic to α (cid:48) i .Using Lemma 6.23 for the second and last estimates we obtain( α i | α i +1 ) ∼ δ ( α (cid:48)(cid:48) i | α (cid:48)(cid:48) i +1 ) ∼ δ ( α (cid:48) i | α (cid:48) i +1 ) x + t = ( α i | α i +1 ) x + t ≥ ( γ | γ ) x + λ + 100 δ + t ∼ δ ( γ | γ ) x + λ + 100 δ, The total errors add up to less than 100 δ , so we obtain( α i | α i +1 ) ≥ ( γ | γ ) x + λ. A similar computation yields, for each i ,( γ | α i ) ≥ ( γ | γ ) − ( R X + 100 δ ) . We have thus verified the hypothesis of Lemma 6.20 in this case, with S = S = R X + 100 δ . Case 2. d ( γ ( t ) , S W ) < R X + 10 δ . OUNDARIES OF DEHN FILLINGS 39
Let γ (cid:48) i be the maximal initial subgeodesic of γ i entirely contained in N = N R + λ +10 δ ( S W ). Recall that by assumption N is isometric to a subspace of T G ,so let us now regard N as a subspace of T G . Since T G is δ –visual, γ (cid:48) i is con-tained in the 2 δ –neighborhood of some ray γ (cid:48)(cid:48) i . There exists a sequence of rays γ (cid:48)(cid:48) = α (cid:48)(cid:48) , . . . , α (cid:48)(cid:48) n = γ (cid:48)(cid:48) , all starting at 1 ∈ G so that ( α (cid:48)(cid:48) i | α (cid:48)(cid:48) i +1 ) ≥ ( γ (cid:48)(cid:48) | γ (cid:48)(cid:48) ) + λ +10 δ and ( α (cid:48)(cid:48) i | γ (cid:48)(cid:48) ) ≥ ( γ (cid:48)(cid:48) | γ (cid:48)(cid:48) ) − R G . Let α (cid:48) i be the maximal initial subgeodesic of α (cid:48)(cid:48) i contained in N . We now switch back to thinking of N as a subspace of T W . Since T W is δ –visual, there exist rays α i , starting at 1, so that α (cid:48) i is contained in the10 δ –neighborhood of α i . We can take α = γ , α n = γ . It is now straightforwardto check that ( α i | α i +1 ) ≥ ( γ | γ ) + λ and ( α i | γ ) ≥ ( γ | γ ) − R G − δ .We have verified the hypothesis of Lemma 6.20 in this case, with S = S = R G + 10 δ .Taking S to be the maximum of S and S , we have verified the hypothesisof Lemma 6.20 in both cases, and conclude using this lemma that ( ∂T W , ρ W ) islinearly connected with constant independent of the spiderweb chosen. (cid:3) Approximating boundaries are spheres
Statement and notation.
In this section we fix ( G, P ) relatively hyperbolicwith P = { P , . . . , P n } where each P i is virtually Z ⊕ Z . We let X be a cuspedspace for the pair and assume that ∂ ( G, P ) = ∂X is a 2–sphere. We also fix a Dehnfilling π : G → G = G ( N , . . . , N n ) so that each N i is isomorphic to Z , and supposethe filling is long enough to apply Theorem 5.2. For θ, δ the constants in Theorem5.2, we consider a θ –spiderweb W associated to this filling (Definition 4.2). Thespiderweb is preserved by a finitely generated free group K W < ker π . We denotethe rank of K W by k . The associated truncated quotient T W (Definition 5.1) is δ –hyperbolic by Theorem 5.2.In this section we describe the Gromov boundary of the truncated quotient: Proposition 7.1.
With the above assumptions, ∂T W is homeomorphic to S . Reduction to a homology computation.
Thanks to the following lemma,the proof of Proposition 7.1 is reduced to a homology computation.
Lemma 7.2.
Let M be a compact Hausdorff space and let S be a dense subset of M homeomorphic to a surface with empty boundary. Suppose that m = M \ S ) is finite and that the dimension of H ( S, Z / is at most max { m − , } . Then M is homeomorphic to S .Proof. When we refer to ‘homology’ in this proof we always mean homology with Z / S is a surface of finite type. Indeed, this followsfrom the fact that surfaces of infinite type have infinite dimensional first homology,as one can deduce from the classification of non-compact surfaces given in [Ric63,Theorem 3].Let p be the number of punctures of S . If p = 0, then M = S is a compactsurface with H ( M ; Z /
2) = 0, so M ∼ = S .Now suppose p >
0, and let S be the closed surface obtained filling in thepunctures of S . Note that S is equal to the end-compactification of S .Since M \ S is finite, S is open and M \ S is totally disconnected. Also, byassumption M is compact and Hausdorff and S is dense in M , and hence the universal property of end-compactifications [Fre31, Satz 6] gives us a map h : S → M restricting to the identity on S . Since S is dense in M the map h is surjective. Inparticular, m ≤ p . Moreover, if d = dim Z / H ( S ; Z /
2) and r = dim Z / H ( S ; Z / r = d + p −
1. By assumption r ≤ m −
1, so we must have d = 0 and thus S ∼ = S . Finally p − m = r + 1 − m ≤ m − − m = 0, again by assumption.This shows that h also restricts to a bijection between S \ S and M \ S , and so h is a homeomorphism. (cid:3) Loops and Cantor sets in disks.
By a
Cantor set we mean a totally discon-nected compact metrizable space with no isolated points. This subsection is aboutCantor sets in the plane or in S , and doesn’t refer directly to our group-theoreticsetup. We will see later that Λ( K W ) is a Cantor set, and use the following lemmasto control how Λ( K W ) sits in ∂X . Lemma 7.3.
Let C be a Cantor set contained in an open disk D . Suppose that { U i } i ∈ I is a finite collection of disjoint clopen subsets of C whose union is C .Then there exists a finite collection of closed subdisks { D i } i ∈ I , so that for alldistinct j, k ∈ I we have D j ∩ D k = ∅ and for all j we have C ∩ ˚ D j = U j .Proof. Let C std be the standard middle-third Cantor set in the plane. It is knownthat any homeomorphism f : C → C std extends to a homeorphism f : D → R (see[Moi77, Chapter 13]). It is then easy to construct a homeomorphism f so that thecollection of clopen sets { f ( U i ) } admits a family of disks in R as in the statement,which can be then pulled back to D using f . (cid:3) Lemma 7.4.
Let C be a Cantor set contained in S . Suppose that U is a clopensubset of C . If D , D are closed disks in D with ˚ D i ∩ C = U , then ∂D ishomologous to ∂D in H ( S \ C ) .Proof. Let h : S → [0 ,
1] be a smooth function which is zero exactly on U . Thenfor a sufficiently small regular value (cid:15) , the set h − [0 , (cid:15) ] is contained in D ∩ D .The 1–manifold h − ( (cid:15) ) is clearly homologous to both ∂D and ∂D . (cid:3) The particular Cantor set.
In this subsection we return to the situationset up in Subsection 7.1 and verify that the limit set Λ( K W ) in ∂X is a Cantor setwhen the rank k ≥
2. We also describe a nice basis for the topology on Λ( K W ).Recall that the group K W is freely generated by parabolic elements a , . . . , a k . Inparticular it is a free group whose Gromov boundary ∂K W can be identified with theset of all infinite freely reduced words in a ± , . . . , a ± k . The collection of quasiconvexsubgroups A = {(cid:104) a (cid:105) , . . . (cid:104) a k (cid:105)} is malnormal in the free group K W , so the pair( K W , A ) is relatively hyperbolic. Its Bowditch boundary ∂ ( K W , A ) is the quotientof ∂K W obtained by identifying the pairs { wa ∞ i , wa −∞ i } for each i and each freelyreduced w . We can choose w not to end with a i or a − i in such a description. (See[Tra13, Theorem 1.1] for the description of the boundary of a relatively hyperbolicpair ( H, Q ) where H is hyperbolic, cf. [Ger12, GP13, MOY12, Man15].) Lemma 7.5.
There is an equivariant homeomorphism ∂ ( K W , A ) → Λ( K W ) .Proof. The spiderweb axioms ((S1)) and ((S3)) from Definition 4.2 imply that K W is relatively quasiconvex in ( G, P ), using [Hru10, Definition 6.5 (QC-3)]. In partic-ular, the limit set Λ( K W ) is equivariantly homeomorphic to the relative boundary OUNDARIES OF DEHN FILLINGS 41 of K W endowed with the peripheral structure induced by G (see e.g. the alter-native definition of relative quasiconvexity [Hru10, Definition 6.2 (QC-1)]), whichcorresponds to the peripheral structure on K W used to define ∂ ( K W , A ). (cid:3) Definition 7.6.
Given a natural number k , an index i ∈ { , . . . , k } , a naturalnumber j and a word w which does not end with a i or a − i , let B ( w, a i , j ) be theimage in ∂ ( K W , A ) of the set of all infinite freely reduced words beginning with wa ji or wa − ji . Lemma 7.7.
The sets B ( w, a i , j ) are clopen in ∂ ( K W , A ) .Proof. The subset A of ∂K W of all infinite (freely reduced) words which start with wa ji or wa − ji is closed, whence compact, and hence so is its image B ( w, a i , j ) in ∂ ( K W , A ). Moreover, A c is also closed and hence so is its image B in ∂ ( K W , A ). Itis easily seen that if the infinite freely reduced word w (cid:48) does not start with either wa ji or wa − ji then no word identified to w (cid:48) in ∂ ( K W , A ) starts with either wa ji or wa − ji , hence B = B ( w, a i , j ) c , and B ( w, a i , j ) is clopen. (cid:3) Corollary 7.8. If k ≥ then ∂ ( K W , A ) is a Cantor set.Proof. The fact that ∂ ( K W , A ) is totally disconnected follows from the fact thatthe B ( w, a i , j ) are clopen. It is also easy to see that it does not have isolatedpoints. Finally, ∂ ( K W , A ) is compact and metrizable since it is (homeomorphic to)the boundary of a proper hyperbolic space. (cid:3) The following two lemmas follow directly from the definitions.
Lemma 7.9.
For any i and any w that does not end with a i or a − i we have w. B (1 , a i ,
1) = B ( w, a i , . Lemma 7.10.
For any w which does not end with a i or a − i and any j > , theset B ( w, a i , j ) is equal to B ( w, a i , minus the union (cid:91) j (cid:48) Let U ⊆ ∂ ( K W , A ) be clopen. Then U is a finite disjoint union ofsets {B ( w s , a i s , j s ) } s ∈ J .Proof. Let U (cid:48) be the preimage of U in ∂K W , and note that U (cid:48) is clopen. Claim. There is an n > v is an infinite freely reduced wordwhich coincides with some w ∈ U (cid:48) on an initial subword of length n , then v ∈ U (cid:48) Proof. Suppose not. Then there is a sequence of pairs { ( v i , w i ) } i ∈ N so that v i coincides with w i on an initial subword of length i , w i ∈ U (cid:48) , but v i / ∈ U (cid:48) . Thecommon prefixes u i subconverge to an infinite word which is in the closure both of U (cid:48) and of its complement, contradicting the fact that U (cid:48) is clopen. (cid:3) Now let J be the set of prefixes of words in U (cid:48) of length n which end with apositive power of one of the generators a i . Any s ∈ J can be written uniquelyas a freely reduced word w s a j s i s . Then {B ( w s , a i s , j s ) } s ∈ J satisfies the requiredproperties. (cid:3) Proof of Proposition 7.1. We use the notation set up at the beginning ofthe section. Note that ∂X \ Λ( K W ) has a K W –action, which makes its homologyinto a K W –module. Recall that K W is free of rank k . Lemma 7.12. As a K W –module, H ( ∂X \ Λ( K W ); Z / has rank at most max { k − , } .Proof. When we refer to ‘homology’ in this proof we always mean homology with Z / k = 0 , k ≥ 2. Let Z = ∂X \ Λ( K W ). We identify ∂ ( K W , A ) with Λ( K W ) (which we can do in view ofLemma 7.5). Recall that Λ( K W ) is a Cantor set by Corollary 7.8. For each set B ( w, a i , j ) as in Definition 7.6, let l w,a i ,j be a loop in ∂X bounding a disk thatintersects Λ( K W ) in B ( w, a i , j ) (which exists by Lemma 7.3 in view of the fact that B ( w, a i , j ) is clopen, see Lemma 7.7). The element of H ( Z ) represented by l w,a i ,j depends only on w, a i , and j , by Lemma 7.4. As a first step in the proof, let usshow that such loops generate H ( Z ). It suffices to prove that any simple loop l in Z is, homologically, a sum of loops l w,a i ,j . This is because it suffices to considersmooth self-transverse loops, and each of those is homologically a sum of simpleloops. Let D be one of the disks in ∂X bounded by l . It follows from Lemma 7.11that D ∩ Λ( K W ) is a disjoint union of sets of the form B ( w, a i , j ), which in turnimplies, in view of Lemma 7.3, that homologically l is a sum of loops l w,a i ,j .We now prove that each loop l w,a i ,j is homologically a sum of loops of the form l w (cid:48) ,a i (cid:48) , . Consider some l w,a i ,j . By Lemma 7.10, one of the disks D bounded by l w,a i , has the property that D ∩ Λ( K W ) is a disjoint union of B ( w, a i , j ) and sets B ( w (cid:48) , a i (cid:48) , l w,a i ,j is homologically a sum of loops l w (cid:48) ,a i (cid:48) , , asrequired.Homologically, each of these loops l w (cid:48) ,a i (cid:48) , is in the K W –orbit of the loop l ,a i (cid:48) , by Lemma 7.9. In particular, the k loops { l ,a i , } generate H ( Z ) as a K W –module.If k ≥ 2, these loops can be chosen to encircle disjoint discs whose union containsΛ( K W ), so we have (cid:80) l (1 , a i , 1) = 0 in H ( S \ Λ( K W )). Any one of the generatorscan be written in terms of the others, so the rank is at most k − (cid:3) Proof of Proposition 7.1. For sufficiently long fillings and any spiderweb W withsuitable parameter, by Theorem 5.2 there is a normal covering map ∂X \ Λ( K W ) → ∂T W \ F with deck group K W , where F is the union of all limit sets of horospherequotients Λ(Σ c /K c ) for c ∈ C ( W ), which in our case is a finite set with 2 k elementsif k is the rank of the free group K W (we assume k ≥ S = ∂T W \F is a 2–manifold, and since ∂T W \ F is open and dense in ∂T W by Theorem 5.2.(3)and ∂T W is compact, in view of Lemma 7.2 we are left to show that the dimensionof H ( S, Z / 2) is at most 2 k − 1. From the short exact sequence1 → π ( ∂X \ Λ( K W )) → π ( S ) → K W → , we see that this dimension is the sum of k and the rank of H ( ∂X \ Λ( K W )) as a K W –module, so we are done by Lemma 7.12. (cid:3) Ruling out the Sierpinski carpet In the next section, we will prove Theorem 1.2 by first showing that the boundary ∂G is planar, using Lemma 3.9 and a criterion of Claytor [Cla34]. A result ofKapovich and Kleiner [KK00, Theorem 4] (along with [GM, Theorem 1.2]) then OUNDARIES OF DEHN FILLINGS 43 implies that ∂G is either S or a Sierpinski carpet. In this section, we rule out thepossibility that it is a Sierpinski carpet. Definition 8.1. Let M be a metric space, let f : S → M be continuous, and let (cid:15) > 0. We say that f has an (cid:15) –filling if there is a triangulation of the unit disk D and a (not necessarily continuous) extension of f to ¯ f : D → M , so that eachsimplex of the triangulation is mapped by ¯ f to a set of diameter at most (cid:15) . Definition 8.2. Say a metric space M is weakly simply connected if, for everycontinuous f : S → M and every (cid:15) > f has an (cid:15) –filling.The following two lemmas are easy. Lemma 8.3. For a compact metrizable space M , being weakly simply connected isindependent of the metric. Lemma 8.4. Any simply connected metric space is weakly simply connected. Lemma 8.5. The Sierpinski carpet (with any metric) is not weakly simply con-nected.Proof. By Lemma 8.3, we just need to check this for a Sierpinski carpet S embeddedin the 2–sphere and endowed with the induced metric. Suppose by contradictionthat S is weakly simply connected. Then it is easily seen that, given any (cid:15) > (cid:96) contained in S , we can find a continuous map f : D → S whoseimage is contained in the (cid:15) –neighborhood of S and so that f ( ∂D ) is (cid:96) . However,the image of a continuous map f : D → S so that f ( ∂D ) is a simple loop (cid:96) contains one of the two connected components of S \ (cid:96) . For (cid:96) a peripheral circleof S and (cid:15) > S \ (cid:96) is containedin the (cid:15) –neighborhood of S , a contradiction. (cid:3) Theorem 8.6. Suppose the compact metric space Z is a weak Gromov–Hausdorfflimit of { Z n } n ∈ N , and suppose that there is some L ≥ so that all the spaces Z n and Z are L –linearly connected. If the spaces Z n are weakly simply connected, thenso is Z .Proof. Assume that { Z n } and Z are as in the hypothesis of the theorem. Then thereare a K ≥ K, (cid:15) n )–quasi-isometries ψ n : Z n → Z and φ n : Z → Z n whichare (cid:15) n –quasi-inverses of one another and for which lim n →∞ (cid:15) n = 0.Let f : S → Z , and let (cid:15) > 0. Choose some (cid:15) (cid:48) < (cid:15) LK , and fix some n so that (cid:15) n < (cid:15) (cid:48) .We want to build an (cid:15) –filling (in Z ) from some (cid:15) (cid:48) –filling in Z n . We’ll firstapproximate f by a discrete map, push it to Z n , fill, and then push the filling backto Z .Let Θ ⊂ S be a discrete set with at least three points. We say that θ, θ (cid:48) ∈ Θare consecutive if they bound a (necessarily unique) interval I =: [ θ, θ (cid:48) ] in S whoseinterior is disjoint from Θ. By refining Θ we can ensure the following: • If θ, θ (cid:48) are consecutive (on S ), and x ∈ [ θ, θ (cid:48) ], then d Z ( f ( θ ) , f ( x )) < (cid:15) (cid:48) .In particular, f ( S ) lies in an (cid:15) (cid:48) –neighborhood of f (Θ), and φ n f ( S ) lies in a( K + 1) (cid:15) (cid:48) –neighborhood of φ n f (Θ). More to the point, if θ, θ (cid:48) are consecutiveelements of Θ, then d Z n ( φ n f ( θ ) , φ n f ( θ (cid:48) )) < ( K + 1) (cid:15) (cid:48) , and so there is an arc in Z n of diameter at most L ( K + 1) (cid:15) (cid:48) joining φ n f ( θ ) to φ n f ( θ (cid:48) ). Concatenating thesearcs, we obtain a continuous f (cid:48) : S → Z n . Claim. For any x ∈ S , we have d Z ( ψ n f (cid:48) ( x ) , f ( x )) < (cid:15) .Assuming the claim, we argue as follows. The map f (cid:48) has an (cid:15) (cid:48) –filling F (cid:48) : D → Z n . Define a filling F : D → Z of f by: F ( x ) = (cid:40) f ( x ) x ∈ S ψ n F (cid:48) ( x ) x ∈ D \ S . Since (cid:15) n < (cid:15) (cid:48) , and since the difference between ψ n f (cid:48) and f is at most (cid:15) on S , F is a (cid:0) ( K + 1) (cid:15) (cid:48) + (cid:15) (cid:1) –filling of f . But (cid:15) (cid:48) < (cid:15) K +1) , so F is an (cid:15) –filling. Modulo theclaim, the theorem is proved. Proof of Claim. Note first that if x ∈ Θ, then f (cid:48) ( x ) = φ n f ( x ), so d Z ( ψ n f (cid:48) ( x ) , f ( x )) ≤ (cid:15) n < (cid:15) (cid:48) < (cid:15) . Suppose now that x / ∈ Θ. There are consecutive θ, θ (cid:48) ∈ Θ so that x lies in [ θ, θ (cid:48) ],and so that d Z ( f ( x ) , f ( θ )) < (cid:15) (cid:48) . It follows that d Z n ( φ n f ( x ) , φ n f ( θ )) < ( K + 1) (cid:15) (cid:48) .From the construction of f (cid:48) we have d Z n ( f (cid:48) ( x ) , f (cid:48) ( θ )) ≤ L ( K + 1) (cid:15) (cid:48) . Pushingback to Z we get d Z ( ψ n f (cid:48) ( x ) , f ( x )) ≤ d Z ( ψ n f (cid:48) ( x ) , ψ n f (cid:48) ( θ )) + d Z ( ψ n f (cid:48) ( θ ) , f ( θ )) + d Z ( f ( θ ) , f ( x )) < K ( L ( K + 1) (cid:15) (cid:48) + (cid:15) (cid:48) ) + (cid:15) (cid:48) + (cid:15) (cid:48) ≤ LK (cid:15) (cid:48) < (cid:15) , and the claim is proved. (cid:3) With the claim proved, the proof of Theorem 8.6 is complete. (cid:3) Proof of Theorem 1.2 In this section we will prove Theorem 1.2, which we restate for the convenienceof the reader. Theorem 1.2. Let G be a group, and P = { P , . . . , P n } a collection of free abeliansubgroups. Suppose that ( G, P ) is relatively hyperbolic, and that ∂ ( G, P ) is a –sphere.Then for all sufficiently long fillings G → G = G ( N , . . . , N n ) with P i /N i virtu-ally infinite cyclic for each i , we have that G is hyperbolic with ∂G homeomorphicto S .Proof. First of all, notice that the P i have rank 2, because they act properly dis-continuously and cocompactly on the complement of the corresponding parabolicpoint in ∂ ( G, P ), which is homeomorphic to R . Suppose that the filling is longenough that Theorem 5.2, Theorem 5.3, Theorem 5.4 and Proposition 7.1, as wellas [GM, Theorem 1.2], all apply. In particular, G is hyperbolic. By Theorem 4.3there exists a sequence of spiderwebs W i and (visual) metrics ρ W i on ∂T W i so that • each ∂T W i is homeomorphic to a 2–sphere (see Proposition 7.1), • there exists L so that each ρ W i is L –linearly connected (see Theorem 5.4), • ( ∂T W i , ρ W i ) weakly Gromov–Hausdorff converges to a (visual) metric on ∂T G (see Theorem 5.3), which is homeomorphic to ∂G . OUNDARIES OF DEHN FILLINGS 45 It follows from [GM, Theorem 1.2] that ∂G is a Peano continuum without localcut points. Moreover, by Lemma 3.9, ∂G does not contain an embedded topolog-ical copy of any non-planar graph and hence, by [Cla34], ∂G is planar. Since, asmentioned, ∂G does not contain local cut points, it must be a sphere or a Sierpinskicarpet, by [KK00, Theorem 4]. The latter is ruled out by Theorem 8.6 and Lemma8.5. (cid:3) Remark 9.1. There is another possible variation of the argument above that doesnot use [GM, Theorem 1.2]. First of all, ∂G is connected because it is a weakGromov–Hausdorff limit of connected spaces. Moreover, it does not have globalcut points because it is the connected boundary of a hyperbolic group [Swa96].Hence, [Cla34] applies and ∂G is planar. At this point we would have to adapt thearguments in Section 8 to deal with a planar continuum properly contained in S .10. Proof of Corollary 1.4 In this section we prove Corollary 1.4, that the Cannon Conjecture implies theRelative Cannon Conjecture. To that end, suppose that the Cannon Conjecture istrue and suppose that ( G, P ) is a relatively hyperbolic pair, where P = { P , . . . , P n } is a finite collection of free abelian subgroups of rank 2 and suppose further thatthe Bowditch boundary of ( G, P ) is homeomorphic to S . Definition 10.1. A sequence { η i : G → G i } of homomorphisms is stably faithful if η i is faithful on the ball of radius i about 1 in G .By choosing fillings kernels K i,j (cid:69) P j where P j /K i,j is infinite cyclic, but theslope is growing, we obtain a stably faithful sequence of fillings G → G i . Thegroups G i are all hyperbolic relative to a finite collection of infinite cyclic groups,and so are in fact hyperbolic. By Theorem 1.2, we may assume that each G i isa hyperbolic group with 2–sphere boundary. By the Cannon Conjecture, each G i admits a discrete faithful representation into Isom( H ). Pre-composing with thequotient maps G → G i , we get a stably faithful sequence of representations of ρ i : G → Isom( H ).There are two cases (after passing to a subsequence of { ρ i } ):(1) Up to conjugacy in Isom( H ), the representations ρ i converge to a repre-sentation ρ ∞ : G → Isom( H ).(2) The representations ρ i diverge in the Isom( H )–character variety of G .Suppose that the first case holds. The ρ i are stably faithful with discrete image. Claim 10.1.1. Suppose that the limiting representation ρ ∞ is not discrete andfaithful. Then for every (cid:15) > { g, h } of noncommuting elementsof G and a point x ∈ H so that ρ ∞ ( g ) and ρ ∞ ( h ) move x distance less than (cid:15) . Proof of Claim 10.1.1. If the limiting representation is not faithful then there arecertainly non-commuting elements g and h in the kernel, which will suffice.On the other hand, suppose that the limiting representation is faithful, butindiscrete. Then there are elements g j of G which are not in the kernel of ρ ∞ butso that ρ ∞ ( g j ) tends towards the identity. Unless the g j eventually commute witheach other, taking two elements far enough along the sequence will suffice for g and h . Thus we may suppose that all of the g j commute with each other, which impliesthat they all preserve some point at infinity, some geodesic or some point in the interior of H . Since ρ ∞ is faithful, it is not elementary, so there is some γ ∈ G sothat ρ ∞ ( γ ) does not preserve this set. We can take g j and g γj for our g and h (forlarge enough j ). (cid:3) The condition from Claim 10.1.1 is an open condition, so for all but finitely many i the elements ρ i ( g ) and ρ i ( h ) move some point in H a distance smaller than (cid:15) .Since for large i the elements ρ i ( g ) and ρ i ( h ) are nontrivial, for small enough (cid:15) thisviolates Margulis’ Lemma and shows that the image ρ i is not discrete, which is acontradiction. This implies that in the first case the representation ρ ∞ is discreteand faithful, so G is Kleinian, as required.It remains to rule out the second case, that the sequence ρ i diverges. If it does,then by choosing basepoints appropriately and rescaling these representations limitto an action of G on an R –tree T with no global fixed point.A standard argument (see, for example, the proof of [GM, Theorem 6.1]) showsthat arc stabilizers for the G –action on T are metabelian and hence small. Sincesmall subgroups of G are finitely generated, this means that arc stabilizers satisfythe ascending chain condition. Therefore, by [BF95, Proposition 3.2.(2)] the actionof G on T is stable. It now follows from [BF95, Theorem 9.5] that G splits over asmall-by-abelian (and hence small) subgroup.However, all small subgroups of G are elementary, but we know that G admitsno elementary splittings (by work of Bowditch [Bow99a, Bow99b, Bow01, Bow12],see [GM, Corollary 7.9]). This implies that G is Kleinian, as required. Appendix A. δ –hyperbolic technicalities In this appendix, we collect some technical results which are needed for the proofsin this paper, but which are proved using standard arguments and are probably wellknown to experts. In Subsection A.1, we prove that the various different approachesto building “cusped” spaces all result in quasi-isometric spaces. The key applicationof this in our paper is Corollary A.9. In Subsection A.2 we collect some results about δ –hyperbolic geometry, which are all well known. The sole innovation is to recordconstants.A.1. Quasi-isometric horoballs and cusped spaces. The combinatorial cuspedspace in Definition 2.8 is one of several ways to build a “cusped space” whose hyper-bolicity detects the relative hyperbolicity of the pair ( G, P ). Each method beginswith a Cayley graph for G , and attaches some kind of “horoball” to each left cosetof an element of P . In [Bow12], Bowditch glues ‘hyperbolic spikes’ (the subset[0 , × [1 , ∞ ) in the upper half-space model of H ) to each edge in the includedcosets of Cayley graphs of the peripheral subgroups, with the lines { } × [1 , ∞ ) and { } × [1 , ∞ ) glued according to when edges share vertices. The resulting cuspedspace is also Gromov hyperbolic if and only if ( G, P ) is relatively hyperbolic, by[Bow12]. Another way of building horoballs on graphs is provided by Cannon andCooper [CC92]. In this case, the horospheres are copies of Γ, but scaled at depth d by λ d for some λ ∈ (0 , 1) (Cannon and Cooper chose λ = e − , which is the mostnatural choice when comparing to the metric in H n ).In order to be able to translate results proved with different cusped spaces tothe other settings, it is convenient to notice that it is not only the case that allof these constructions provide characterizations of relative hyperbolicity, but thatthey provide G –equivariantly quasi-isometric cusped spaces, a fact well-known to OUNDARIES OF DEHN FILLINGS 47 experts. This is what we explain in this subsection. Throughout the paper, the factthat we can use results from the literature proved using different models is justifiedby the results in this section.We consider three types of horoballs, each depending on some scaling factor λ . Definition A.1 (Combinatorial Horoball) . Let Γ be a graph and λ > combinatorial horoball based on Γ with scaling factor λ is the graph as definedin Definition 2.7 except that horizontal edges are added between ( v, k ) and ( w, k )when 0 < d Γ ( v, w ) ≤ λ k . We denote this space by CH (Γ , λ ).Note that Groves and Manning used λ = 2, as in Definition 2.7. Definition A.2 (Cannon–Cooper Horoball) . Let Γ be a metric graph and let λ > CC-horoball based on Γ with scaling factor λ to be a metric graph H (Γ)whose vertex set is Γ (0) × Z ≥ , and with two types of edges:(1) A vertical edge of length 1 from ( v, n ) to ( v, n + 1) for any v ∈ Γ (0) andany n ≥ (cid:15) is an edge of length l in Γ joining v to w , and n ≥ 0, there is a horizontal edge of length λ − n l joining ( v, n ) to ( w, n ).We denote this space by CCH (Γ , λ ).Note that Cannon and Cooper used λ = e .To define the Bowditch horoball, it is convenient to use the notion of a warpedproduct of length spaces. Definition A.3 (Warped product of length spaces) . [Che99] Let ( B, d B ) (the base )and ( F, d F ) (the fiber ) be two length spaces, and let f : B → [0 , ∞ ) be a continuousfunction (the warping function ). Let t (cid:55)→ ( β ( t ) , γ ( t )) define a path σ : [0 , → B × F .Define a length by first considering, for each partition τ = { t < t < · · · Bowditch Horoball based on Γ with scaling factor λ is thewarped product [0 , ∞ ) × λ − t Γ . We denote this space by BH (Γ , λ ).Note that Bowditch used λ = e .The following result is elementary and probably well known to many experts.The proof is very similar to part of the proof of [CC92, Theorem, § . 2] (see also[Dur14, Proposition 3.2] for more details). Neither Cannon and Cooper nor Durhamdeal with a general graph, but this is irrelevant for the proofs. We leave the detailsas an exercise for the reader. Proposition A.5. Suppose that λ , λ > are constants and that Γ is a graph.The map Γ (0) × Z ≥ → [0 , ∞ ) × λ − t Γ defined by ( v, n ) (cid:55)→ (cid:18) ln( λ )ln( λ ) n, v (cid:19) , extends naturally to quasi-isometries CH (Γ , λ ) → BH (Γ , λ ) , and CCH (Γ , λ ) → BH (Γ , λ ) , by mapping edges in the left-hand spaces to geodesics in BH (Γ , λ ) . Remark A.6. If some care is not taken then different kinds of horoballs may notbe quasi-isometric. For example, if the warping function f for [0 , ∞ ) × f Γ is takento be doubly-exponential, the resulting horoball will still be Gromov hyperbolic,but will not be quasi-isometric to BH (Γ , λ ).From each kind of horoball, there is then an associated cusped space , obtainedfrom the Cayley graph of G by gluing the horoballs based on the Cayley graphs of P ∈ P onto the cosets in the same way is described in Definition 2.8 above.The following is a straightforward application of Proposition A.5. Corollary A.7. Suppose that G is a group, and that P is finite collection of finitelygenerated subgroups of G . The three kinds of cusped spaces obtained by gluing eithercombinatorial, Cannon–Cooper, or Bowditch horoballs to the Cayley graph of G areall quasi-isometric via maps extending the identity map on the Cayley graph of G .In particular, any one of them is Gromov hyperbolic if and only if any of theothers is. Quasi-isometric proper Gromov hyperbolic spaces have quasi-symmetric bound-aries (see for example [BS07, Theorem 5.2.17]). Corollary A.8. Suppose that ( G, P ) is relatively hyperbolic. Let X CH , X CCH and X BH be the three cusped spaces for ( G, P ) associated to the three kinds of horoballs.Equip the (Bowditch) boundaries ∂X CH , ∂X CCH and ∂X BH with visual metricsbased at . These boundaries are quasi-symmetric. By Lemma 2.6, either all these boundaries are linearly connected, or none ofthem are. The following is now an immediate consequence of [MS11, Proposition4.10] and results of Bowditch ([Bow99a, Bow99b, Bow01], see [GM, Theorem 7.3],for example). Corollary A.9. Suppose that ( G, P ) is relatively hyperbolic, that P consists offinitely presented groups with no infinite torsion subgroups, and that ( G, P ) has nonontrivial peripheral splittings. Then the boundary of the cusped space of ( G, P ) (with respect to any type of horoball) is linearly connected. A.2. δ –hyperbolic geometry. All lemmas in this section are well-known factsabout δ –hyperbolic spaces. We include proofs for completeness and to explicitlykeep track of how the constants appearing in the construction of spiderwebs dependon δ . Lemma A.10. Let Y be a δ –hyperbolic space and let A ⊆ Y be any set. Then theunion Z of all geodesics connecting pairs of points in A is δ –quasiconvex. OUNDARIES OF DEHN FILLINGS 49 Proof. For i = 1 , 2, let z i ∈ [ x i , y i ] for some x i , y i ∈ A , and pick any z ∈ [ z , z ].To prove 2 δ –quasiconvexity we can just notice that z is 2 δ –close to either [ x , z ] ⊆ [ x , y ], [ x , x ] or [ x , z ] ⊆ [ x , y ], and all such geodesics are contained in Z . (cid:3) Definition A.11. For a set W , denote the power set of W by 2 W . Suppose that X is a metric space and W ⊂ X . Let π W : X → W be closest point projection, so π W ( x ) is the set of all x (cid:48) ∈ W satisfying d X ( x, x (cid:48) ) = d X ( x, W ). Lemma A.12. Let Y be a δ –hyperbolic space and let W ⊆ Y be Q –quasiconvex.Let x, y ∈ X , and let x (cid:48) ∈ π W ( x ) and y (cid:48) ∈ π W ( y ) . If d Y ( x (cid:48) , y (cid:48) ) > δ + 2 Q then d Y ([ x, y ] , W ) ≤ δ + Q and max { d Y ([ x, y ] , x (cid:48) ) , d Y ([ x, y ] , y (cid:48) ) } ≤ δ + 2 Q .Proof. Pick any point p on a geodesic [ x (cid:48) , y (cid:48) ] satisfying d Y ( p, x (cid:48) ) , d Y ( p, y (cid:48) ) > δ + Q .The quasi-convexity of W ensures that d ( p, W ) ≤ Q . Slimness of geodesic quadri-laterals implies that p is 2 δ –close to some point q lying on a geodesic [ x, x (cid:48) ], [ y, y (cid:48) ]or [ x, y ]. We will rule out the first two possibilities, and deduce that d ([ x, y ] , p ) ≤ δ and so d ([ x, y ] , W ) ≤ δ + Q .By symmetry, we can assume q ∈ [ x, x (cid:48) ]. Since d Y ( q, x (cid:48) ) ≥ d Y ( p, x (cid:48) ) − δ > δ + Q and d Y ( x, q ) = d Y ( x, x (cid:48) ) − d Y ( q, x (cid:48) ), we have d Y ( x, W ) ≤ d Y ( x, q ) + d Y ( q, W ) < (cid:16) d Y ( x, x (cid:48) ) − δ − Q (cid:17) + 2 δ + Q = d Y ( x, x (cid:48) ) , contradicting the fact that x (cid:48) ∈ π W ( x ).To obtain the second assertion, note that we could have chosen p at distance4 δ + Q + (cid:15) from x (cid:48) or y (cid:48) for any sufficiently small (cid:15) > (cid:3) Lemma A.13. Let Y be a δ –hyperbolic space and let W , W be Q –quasiconvexsubsets of Y . Also, let γ be any geodesic from some point p ∈ W to some point p ∈ W . Then any geodesic α from W to W is contained in N Q +2 δ ( W ) ∪ N Q +2 δ ( W ) ∪ N δ ( γ ) .Proof. Let q i be the endpoints of α , with q i ∈ W i . Then any point in α is 2 δ –closeto either γ or to a geodesic [ p i , q i ] for some i , and each such geodesic is containedin N Q ( W i ), so we are done. (cid:3) Definition A.14. We say that a path α in a geodesic space is C –tight if it is(1) (1 , C )–quasi-geodesic, and(2) for any s ≤ t ≤ u in the domain of α any geodesic from α ( s ) to α ( u ) passes C –close to α ( t ).The path is λ –locally C –tight if α | I is C –tight for every interval I of length at most λ . We remark that in this paper we consider a ( λ, (cid:15) )–quasi-geodesic to be a unitspeed path σ so that d ( σ ( s ) , σ ( t )) ≥ λ − | s − t | − (cid:15) , for all s, t . The results we proveare also true if instead one considers quasi-isometric embeddings of an interval, butall the quasi-geodesics we need are continuous unit speed maps. Lemma A.15. Let Y be a δ –hyperbolic space, let W ⊆ Y be Q –quasiconvex andlet x ∈ X . If w ∈ π W ( x ) , then for any w (cid:48) ∈ W the concatenation of geodesics [ x, w ] , [ w, w (cid:48) ] is (4 δ + 2 Q ) –tight.Proof. We prove that the concatenation satisfies the second condition in the defi-nition of C –tight, with C = 2 δ + Q . We then note that such a concatenation mustbe (1 , C )–quasigeodesic. We can restrict to considering geodesics connecting some z ∈ [ x, w ] to some z ∈ [ w, w (cid:48) ]. Choose [ z , w ] ⊆ [ x, w ] and [ w, z ] ⊆ [ w, w (cid:48) ], and let p ∈ [ z , z ], p ∈ [ z , w ], p ∈ [ w, z ] be the internal points of the triangle [ z , z ] ∪ [ z , w ] ∪ [ w, z ].Since p lies on [ w, w (cid:48) ], we have d ( p , W ) ≤ Q , and so d ( p , W ) ≤ δ + Q . Since p lies on a shortest path from x to W , we have d ( p , w ) ≤ δ + Q . Moreover d ( p , w ) = d ( p , w ) ≤ δ + Q . It follows that any point on [ z , w ] ∪ [ z , w ] lies within2 δ + Q of some point on [ z , z ], as required. (cid:3) The proof of the following result is a minor variation of the one from [BH99,III.H.1.13] Lemma A.16. Let Y be a δ –hyperbolic space and let α be a (6 C + 8 δ + 1) –local C –tight path. Then the Hausdorff distance between α and any geodesic with thesame endpoints as α is at most C + 4 δ .Proof. Let us first show α ⊆ N C +2 δ ( γ ), where γ is a geodesic with the same end-points.Let [0 , a ] be the domain of α . Let p = α ( t ) be any point on α at maximal distancefrom γ , and let d = d Y ( α ( t ) , γ ). Suppose by contradiction that d > C + 4 δ . Let R = 3 C + 4 δ + 1 / 2, so that α is 2 R –locally C –tight. Using local C –tightness,there are points x = α ( t ) and y = α ( t ) satisfying t ∈ ( t − R, t ) , t ∈ ( t, t + R )and so that min { d Y ( x, p ) , d Y ( y, p ) } > C + 4 δ . Choose a geodesic [ x, y ]. By localtightness, there is a p (cid:48) ∈ [ x, y ] within C of p .Let x (cid:48) , y (cid:48) ∈ γ be chosen so that d Y ( x, x (cid:48) ) and d Y ( y, y (cid:48) ) are minimal, and let[ x (cid:48) , y (cid:48) ] be the subsegment of γ joining them. Consider a geodesic quadrilateral[ x, y ] ∪ [ x (cid:48) , y (cid:48) ] ∪ [ x, x (cid:48) ] ∪ [ y, y (cid:48) ]. The point p (cid:48) is within 2 δ of some p (cid:48)(cid:48) in one ofthe other three sides. It cannot be [ x (cid:48) , y (cid:48) ], or we would have d = d Y ( p, γ ) ≤ d Y ( p, p (cid:48)(cid:48) ) ≤ C + 2 δ . Suppose on the other hand that p (cid:48)(cid:48) ∈ [ x, x (cid:48) ] (the argument for[ y, y (cid:48) ] is identical). Then d Y ( p, γ ) ≤ C + 2 δ + d − d Y ( p (cid:48)(cid:48) , x ). 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