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Cannon-Thurston Maps for Surface Groups I:Amalgamation Geometry and Split Geometry Mahan Mj.RKM Vidyamandira and RKMVERI,Belur Math,WB-711202,India email:brmahan@http://www.wendangku.net/doc/31f54a976bec0975f465e2bf.html Abstract We introduce the notion of manifolds of amalgamation geometry and its generalisation,split geometry.We show that the limit set of any surface group of split geometry is locally connected,by constructing a natural Cannon-Thurston map.AMS Subject Classi?cation:57M50Contents 1Introduction 31.1Statement of Results .......................31.2History and Present State of the Problem ...........41.3Scheme and Outline of the Paper ................62Preliminaries and Amalgamation Geometry 82.1Hyperbolic Metric Spaces ....................82.2Amalgamation Geometry .. (9)

3Relative Hyperbolicity 14

3.1Electric Geometry (14)

3.2Electric isometries (19)

3.3Nearest-point Projections (20)

3.4Coboundedness and Consequences (22)

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4Universal Covers of Building Blocks and Electric Geometry23

4.1Graph Model of Building Blocks (23)

4.2Construction of Quasiconvex Sets for Building Blocks (25)

5Construction of Quasiconvex Sets and Quasigeodesics28

5.1Construction of BλandΠλ (28)

5.2Heights of Blocks (31)

5.3Admissible Paths (33)

5.4Joining the Dots (36)

5.5Admissible Quasigeodesics and Electro-ambient Quasigeodesics39 6Cannon-Thurston Maps for Surfaces Without Punctures41

6.1Electric Geometry Revisited (41)

6.2Proof of Theorem (42)

7Modi?cations for Surfaces with Punctures42

7.1Partial Electrocution (42)

7.2Amalgamated Geometry for Surfaces with Punctures (45)

8Weakening the Hypothesis I:Graph Quasiconvexity and Graph Amalgamation Geometry48 9Weakening the Hypothesis II:Split Geometry53

9.1More Margulis Tubes in a Block (53)

9.2Motivation for Split Geometry (53)

9.3De?nitions (55)

9.4The Cannon-Thurston Property for Manifolds of Split Geom-

etry (61)

10Generalisation:Incompressible away from Cusps65 11The Minsky Model and Split Geometry:A Sketch66 12Extending the Sullivan-McMullen Dictionary69

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1Introduction

1.1Statement of Results

In this paper and its successor[31],we continue our study of Cannon-Thurston maps and limit sets of Kleinian groups initiated in[28],[29]and

[30].Several questions and conjectures have been made in this context by

di?erent authors:

?1In Section6of[10],Cannon and Thurston raise the following problem:

Question:Suppose a closed surface groupπ1(S)acts freely and properly

discontinuously on H3by isometries.Does the inclusion?i: S→H3extend continuously to the boundary?

The authors of[10]point out that for a simply degenerate group,this is

equivalent to asking if the limit set is locally connected.

?2In[21],McMullen makes the following more general conjecture:

Conjecture:For any hyperbolic3-manifold N with?nitely generated fun-

damental group,there exists a continuous,π1(N)-equivariant map

F:?π1(N)→Λ?S2∞

where the boundary?π1(N)is constructed by scaling the metric on the Cayley graph ofπ1(N)by the conformal factor of d(e,x)?2,then taking the metric completion.(cf.Floyd[13])

?3The author raised the following question in his thesis[26](see also[1]):

Question:Let G be a hyperbolic group in the sense of Gromov acting freely

and properly discontinuously by isometries on a hyperbolic metric space X. Does the inclusion of the Cayley graph i:ΓG→X extend continuously to the(Gromov)compacti?cations?

A similar question may be asked for relatively hyperbolic groups(in the

sense of Gromov[15]and Farb[12]).

The question for relatively hyperbolic groups uni?es all the above ques-

tions and conjectures.

In this paper we introduce the notion of what we call amalgamation

geometry which is,in a way,a considerable generalisation of the notion of

i-bounded geometry introduced in[30].We then generalise it by weakening

the hypothesis to the notion of split geometry.A crucial step in this paper

is to prove:

Theorems9.2and9.3:Letρ:π1(S)→P SL2(C)be a faithful represen-

tation of a surface group with or without punctures,and without accidental

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parabolics.Let M=H3/ρ(π1(S))be of split geometry.Let i be an embed-ding of S in M that induces a homotopy equivalence.Then the embedding ?i: S→ M=H3extends continuously to a map?i:D2→D3.Further,the

limit set ofρ(π1(S))is locally connected.

In fact our methods prove the following considerably stronger result by combining the techniques of this paper with those of[28]and[29].This is a partial a?rmation of McMullen’s conjecture above.

Theorem10.1:Suppose that N h∈H(M,P)is a hyperbolic structure of split geometry on a pared manifold(M,P)with incompressible boundary ?0M.Let M gf denote a geometrically?nite hyperbolic structure adapted to (M,P).Then the map i: M gf→ N h extends continuously to the boundary ?i: M gf→ N h.IfΛdenotes the limit set of M,thenΛis locally connected.

In[31],we shall show that the Minsky model is of split http://www.wendangku.net/doc/31f54a976bec0975f465e2bf.html-bining this with Theorems9.2and9.3,we shall get

Theorem[31]:Letρbe a representation of a surface group H(correspond-ing to the surface S)into P Sl2(C)without accidental parabolics.Let M denote the(convex core of)H3/ρ(H).Further suppose that i:S→M, taking parabolic to parabolics,induces a homotopy equivalence.Then the inclusion?i: S→ M extends continuously to a map?i: S→ M.Hence the limit set of S is locally connected.

Again,combining the Minsky model with Theorem10.1,we shall get Theorem[31]:Suppose that N h∈H(M,P)is a hyperbolic structure on a pared manifold(M,P)with incompressible boundary?0M.Let M gf denote a geometrically?nite hyperbolic structure adapted to(M,P).Then the map ?i: M gf→ N h extends continuously to the boundary?i: M gf→ N h.IfΛdenotes the limit set of M,thenΛis locally connected.

1.2History and Present State of the Problem

The?rst major result that started this entire program was Cannon and Thurston’s result[10]for hyperbolic3-manifolds?bering over the circle with ?ber a closed surface group.

This was generalised by Minsky who proved the Cannon-Thurston result for bounded geometry Kleinian closed surface groups[23].

An alternate approach(purely in terms of coarse geometry ignoring all local information)was given by the author in[28]generalising the results

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of both Cannon-Thurston and Minsky.We proved the Cannon-Thurston result for hyperbolic3-manifolds of bounded geometry without parabolics and with freely indecomposable fundamental group.A di?erent approach based on Minsky’s work was given by Klarreich[19].

Bowditch[3][4]proved the Cannon-Thurston result for punctured sur-face Kleinian groups of bounded geometry.In[29]we gave an alternate proof of Bowditch’s results and simultaneously generalised the results of Cannon-Thurston,Minsky,Bowditch,and those of[28]to all3manifolds of bounded geometry whose cores are incompressible away from cusps.The proof has the advantage that it reduces to a proof for manifolds without parabolics when the3manifold in question has freely indecomposable fundamental group and no accidental parabolics.

McMullen[21]proved the Cannon-Thurston result for punctured torus groups,using Minsky’s model for these groups[24].In[30]we identi?ed a large-scale coarse geometric structure involved in the Minsky model for punctured torus groups(and called it i-bounded geometry).i-bounded geometry can roughly be regarded as that geometry of ends where the bound-ary torii of Margulis tubes have uniformly bounded diameter.We gave a proof for models of i-bounded geometry.In combination with the methods of[29]this was enough to bring under the same umbrella all known results on Cannon-Thurston maps for3manifolds whose cores are incompressible away from cusps.In particular,when(M,P)is the pair S×I,δS×I,for S a punctured torus or four-holed sphere,we gave an alternate proof of McMullen’s result[21].

In this paper,we de?ne amalgamation geometry and prove the Cannon-Thurston result for models of amalgamation geometry.We then weaken this assumption to what we call split geometry and prove the Cannon-Thurston property for such geometries.In[31]we shall show that the Minsky model for general simply or totally degenerate surface groups[25][8]gives rise to a model of split geometry.This will allow us to conclude that all surface groups have the Cannon-Thurston property and hence have locally connected limit sets.In the sequel to this paper[31],we show that the Minsky model for surface groups has split geometry.This proves that surface groups(and more generally Kleinian groups corresponding to manifolds whose cores are incompressible away from cusps)have locally-connected limit sets.

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1.3Scheme and Outline of the Paper

We?rst describe in brief,the philosophy of the proof.Given a simply

degenerate surface(S)group(without accidental parabolics),Thurston[35]

proves that a unique ending laminationλexists.Let M=H3/ρ(π1(S)).In

this situation,it follows from[35]that any sequence of simple closed curves

σi,whose geodesic realizations exit the end,converges toλ.Dual toλ,there

exists an R-tree T and a free action ofπ1(S)on T.Now,eachσi gives rise

to a splitting ofπ1(S),and hence an action ofπ1(S)on a simplicial tree T i.

The sequence of these actions converges to the action ofπ1(S)on T dual to

λ(see for instance,[32]).

The guiding motif of this paper is to?nd geometric realizations of this

sequence of splittings in terms of contiguous blocks B i(each homeomorphic

to S×I).By a geometric realization of a splitting we mean the following:

Margulis tubes T i are chosen,exiting the end of M.Letσi denote the core

geodesic of T i.We require that T i splits some block B i,i.e.B i\T i is

homeomorphic to(S\A(σi))×I,where A(σi)is an annular neighborhood

of a geodesic representative ofσi on S.We require further control on the

geometry of the complementary pieces(S\A(σi))×I.

For conceptual simplicity,assume the T i’s are separating.Di?erent de-

grees of control on the geometry of the pieces(S\A(σi))×I give rise to

di?erent geometries.Fix a piece of(S\A(σi))×I and call it K.It is better to look at the universal cover B i and a lift K? B i.We adjoin the lifts of T i bounding K to K and call it K1.K1shall be referred to as a component of the relevant geometry.

1)Amalgamation Geometry:The simplest geometry arising from this

situation is the case where all K1’s are uniformly quasiconvex in the hyper-

bolic metric on M.This is called amalgamation geometry,and can in brief e described as the geometry in which all components are uniformly(hyper-bolically)quasiconvex.

2)Graph Amalgamation Geometry:Amalgamation geometry is too

restrictive.As a?rst step towards relaxing this hypothesis,we do not de-

mand that the convex hulls CH(K1)’s be contained in uniformly bounded neighborhoods of the respective K1’s in the hyperbolic metric.Instead we construct an auxiliary metric called the graph-metric.Roughly speaking, the graph-metric is the natural simplicial metric on the nerve of the covering of M by the components K1.Graph Amalgamation Geometry is the condi-tion that the convex hulls CH(K1)’s lie in uniformly bounded neighborhood of K1’s in the graph metric.

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3)Split Geometry:So far,we have assumed that each Margulis tube T i is contained wholly in a block B i,splitting it.However,as was pointed out to the author by Yair Minsky and Dick Canary,this is not the most general situation.The T i’s may interlock.To take care of this situation,we allow each tube T i to cut through(partly or wholly)a uniformly bounded number of blocks.The notions of complementary components and the graph metric still make sense.The rsulting geometry is termed split geometry.

We shall take one step at a time in this paper,relaxing the hypothesis in the order above.The additional arguments to be introduced as we proceed from one geometry to the next(more general)one will be described as modi?cations of the core argument relevant to amalgamation geometry.

In the sequel[31],we shall show that simply degenerate ends of hyper-bolic3-manifolds enjoy split geometry.

Outline:A brief outline of the paper follows.Section2deals with pre-liminaries.We also de?ne amalgamation geometry via the construction of a model manifold.

Section3deals with relative hyperbolicity a la Gromov[15],Farb[12] and Bowditch[2].

As in[27],[28],[29],[30],a crucial part of our proof proceeds by con-structing a ladder-like set Bλ? M from a geodesic segmentλ? S and then a retractionΠλof M onto Bλ.

In Section4,we construct a model geometry for the universal covers of building blocks and the relevant geometries(electric and graph models)that will concern us.

We also construct the paths that go to build up the ladder-like set Bλ. We further construct the restriction of the retractionΠλto blocks and show that the retraction does not increase distances much.

In Section5,we put the blocks and retractions together(by adding them one on top of another)to build the ladder-like Bλand prove the main technical theorem-the existence of of a retractΠλof M onto Bλ.This shows that Bλis quasiconvex in M equipped with a model pseudometric.

In Section6,we put together the ingredients from Sections2,3,4and5 to prove the existence of a Cannon-Thurston map for simply or doubly de-generate Kleinian groups corresponding to representations of closed surface groups that have amalgamation geometry.

In Section7,we extend these results to include surface groups with punctures.

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In Section8,we weaken the hypothesis of amalgamation geometry to

what we have called graph amalgamation geometry and describe the modi?-cations necessary to extend our results to such geometries.

In Section9,we weaken the hypothesis further to split geometry which

allows for Margulis tubes to cut across the blocks.

In Section10,we further generalise these result to include hyperbolic

manifolds whose cores are incompressible away from cusps.(We had termed

such manifolds pared manifolds with incompressible boundary in[29].)

In Section11,we give a scheme for proving that the Minsky model for

surface groups[25]has split geometry.Details will appear in the second

part of this paper[31].

In Section12,we propose an extension of the Sullivan-McMullen dic-

tionary between Kleinian groups and complex dynamics,and suggest an analogue of Yoccoz puzzles in the3dimensional setting. Acknowledgements:Its a pleasure to thank Je?Brock,Dick Canary

and Yair Minsky for their support,both personal and mathematical,during

the course of this work.In particular,the generalisations of amalgamation

geometry to graph amalgamation geometry and split geometry were made

to?ll a gap in a previous version of this paper.The gap was brought to my

notice by Minsky and Canary.I,nevertheless,claim credit for any errors

and gaps that might still persist.

2Preliminaries and Amalgamation Geometry

2.1Hyperbolic Metric Spaces

We start o?with some preliminaries about hyperbolic metric spaces in the

sense of Gromov[15].For details,see[11],[14].Let(X,d)be a hyper-

bolic metric space.The Gromov boundary of X,denoted by?X,is

the collection of equivalence classes of geodesic rays r:[0,∞)→Γwith

r(0)=x0for some?xed x0∈X,where rays r1and r2are equivalent if

sup{d(r1(t),r2(t))}<∞.Let X=X∪?X denote the natural compacti?ca-tion of X topologized the usual way(cf.[14]pg.124).

De?nitions:A subset Z of X is said to be k-quasiconvex if any

geodesic joining points of Z lies in a k-neighborhood of Z.A subset Z is

quasiconvex if it is k-quasiconvex for some k.(For simply connected real

hyperbolic manifolds this is equivalent to saying that the convex hull of the

set Z lies in a bounded neighborhood of Z.We shall have occasion to use

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this alternate characterisation.)A map f from one metric space(Y,d Y) into another metric space(Z,d Z)is said to be a(K,?)-quasi-isometric embedding if

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The Amalgamated Building Block

For the construction of an amalgamated block B,I will denote the closed interval[0,3].We will describe a geometry on S×I.B has a geometric core K with bounded geometry boundary and a preferred geodesicγ(=γB) of bounded length.

There will exist?0,?1,D(independent of the block B)such that the following hold:

1.B is identi?ed with S×I

2.B has a geometric core K identi?ed with S×[1,2].(K,in its

intrinsic path metric,may be thought of,for convenience,as a convex hyperbolic manifold with boundary consisting of pleated surfaces.But we will have occasion to use geometries that are only quasi-isometric to such geometries when lifted to universal covers.As of now,we do not impose any further restriction on the geometry of K.)

3.γis homotopic to a simple closed curve on S×{i}for any i∈I

4.γis small,i.e.the length ofγis bounded above by?0

5.The intrinsic metric on S×i(for i=1,2)has bounded geometry,

i.e.any closed geodesic on S×{i}has length bounded below by?1.

Further,the diameter of S×{i}is bounded above by D.(The lat-ter restriction would have followed naturally had we assumed that the curvature of S×{i}is hyperbolic or at least pinched negative.)

6.There exists a regular neighborhood N k(γ)?K ofγwhich is homeo-

morphic to a solid torus,such that N k(γ)∩S×{i}is homeomorphic to an open annulus for i=1,2.We shall have occasion to denote N k(γ) by Tγand call it the Margulis tube corresponding toγ.

7.S×[0,1]and S×[1,2]are given the product structures corresponding

to the bounded geometry structures on S×{i},for i=1,2respectively.

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We next describe the geometry of the geometric core K.K?Tγhas one or two components according asγdoes not or does separate S.These components shall be called amalgamation components of K.Let K1 denote such an amalgamation component.Then a lift K1of K1to K is bounded by lifts Tγof Tγ.The union of such a lift K1along with the lifts Tγthat bound it will be called an amalgamation component of K.

Note that two amalgamation components of K,if they intersect,shall do so along a lift Tγof Tγ.In this case,they shall be referred to as adjacent amalgamation components.

In addition to the above structure of B,we require in addition that there exists C>0(independent of B)such that

?Each amalgamation component of K is C-quasiconvex in the intrinsic metric on K.

Note1:Quasiconvexity of an amalgamation component follows from the fact that any geometric subgroup of in?nite index in a surface group is qua-siconvex in the latter.The restriction above is therefore to ensure uniform quasiconvexity.We shall strengthen this restriction further when we de-scribe the geometry of M,where M is a3-manifold built up of blocks of amalgamation geometry and those of bounded geometry by gluing them end to end.We shall require that each amalgamation component is uniformly quasiconvex in M rather than just in K.

Note2:So far,the restrictions on K are quite mild.There are really two restrictions.One is the existence of a bounded length simple closed geodesic whose regular neighborhood intersects the bounding surfaces of K in an-nulii.The second restriction is that the two boundary surfaces of K have bounded geometry.

The copy of S×I thus obtained,with the restrictions above,will be called a building block of amalgamated geometry or an amalgama-tion geometry building block,or simply an amalgamation block.

Thick Block

Fix constants D,?and letμ=[p,q]be an?-thick Teichmuller geodesic of length less than D.μis?-thick means that for any x∈μand any closed geodesicηin the hyperbolic surface S x over x,the length ofηis greater than ?.Now let B denote the universal curve overμreparametrized such that the length ofμis covered in unit time.Thus B=S×[0,1]topologically.

B is given the path metric and is called a thick building block.

Note that after acting by an element of the mapping class group,we might as well assume thatμlies in some given compact region of Teichmuller

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space.This is because the marking on S×{0}is not important,but rather its position relative to S×{1}Further,since we shall be constructing models only upto quasi-isometry,we might as well assume that S×{0}and S×{1} lie in the orbit under the mapping class group of some?xed base surface. Henceμcan be further simpli?ed to be a Teichmuller geodesic joining a pair (p,q)amongst a?nite set of points in the orbit of a?xed hyperbolic surface S.

The Model Manifold

Note that the boundary of an amalgamation block B i consists of S×{0,3} and the intrinsic path metric on each such S×{0}or S×{3}is of bounded geometry.Also,the boundary of a thick block B consists of S×{0,1},where S0,S1lie in some given bounded region of Teichmuller space.The intrinsic path metrics on each such S×{0}or S×{1}is the path metric on S.

The model manifold of amalgamation geometry is obtained from S×J (where J is a sub-interval of R,which may be semi-in?nite or bi-in?nite.In the former case,we choose the usual normalisation J=[0,∞))by?rst choosing a sequence of blocks B i(thick or amalgamated)and corresponding intervals I i=[0,1]or[0,3]according as B i is thick or amalgamated.The metric on S×I i is then declared to be that on the building block B i.Im-plicitly,we are requiring that the surfaces along which gluing occurs have the same metric.Thus we have,

De?nition:A manifold M homeormorphic to S×J,where J=[0,∞)or J=(?∞,∞),is said to be a model of amalgamation geometry if 1.there is a?ber preserving homeomorphism from M to S×J that lifts

to a quasi-isometry of universal covers

2.there exists a sequence I i of intervals(with disjoint interiors)and

blocks B i where the metric on S×I i is the same as that on some building block B i

3.

i

I i=J

4.There exists C>0such that for all amalgamated blocks B and ge-ometric cores K?B,all amalgamation components of K are C-quasiconvex in M

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Note:The last restriction(4)above is a global restriction on the geom-etry of amalgamation components,not just a local one(i.e.quasiconvexity in M rather than B is required.)

The?gure below illustrates schematically what the model looks like. Filled squares correspond to solid torii along which amalgamation occurs. The adjoining piece(s)denote amalgamation blocks of K.The blocks which have no?lled squares are the thick blocks and those with?lled squares are the amalgamated blocks

Figure1:Model of amalgamated geometry(schematic)

3Relative Hyperbolicity

In this section,we shall recall?rst certain notions of relative hyperbolicity

due to Farb[12],Klarreich[19]and the author[30].Using these,we shall derive certain Lemmas that will be useful in studying the geometry of the

universal covers of building blocks.

3.1Electric Geometry

We start with a surface S(assumed hyperbolic for the time being)of(K,?)

bounded geometry,i.e.S has diameter bounded by K and injectivity radius

bounded below by?.Letσbe a simple closed geodesic on S.Replaceσby a copy ofσ×[0,1],by cutting open alongσand gluing in a copy of

σ×[0,1]=Aσ.(This is like‘grafting’but we shall not have much use for

this similarity in this paper.)Let S G denote the grafted surface.S G?Aσ

has one or two components according asσdoes not or does separate S.Call

these amalgamation component(s)of S We shall denote amalgamation

components as S A.We construct a pseudometric on S G,by declaring the

metric on each amalgamation component to be zero and to be the product

metric on Aσ.Thus we de?ne:

?the length of any path that lies in the interior of an amalgamation com-

ponent to be zero

?the length of any path that lies in Aσto be its(Euclidean)length in the

path metric on Aσ

?the length of any other path to be the sum of lengths of pieces of the above

two kinds.

This allows us to de?ne distances by taking the in?mum of lengths of

paths joining pairs of points and gives us a path pseudometric,which we

call the electric metric on S G.The electric metric also allows us to de?ne

geodesics.Let us call S G equipped with the above pseudometric(S Gel,d Gel)

(to be distinguished from a‘dual’construction of an electric metric S el used

in[30],where the geodesicσ,rather than its complementary component(s)

is electrocuted.)

Important Note:We may and shall regard S as a graph of groups

with vertex group(s)the subgroup(s)corresponding to amalgamation com-ponent(s)and edge group Z,the fundamental group of Aσ.Then S equipped with the lift of the above pseudometric is quasi-isometric to the tree corre-

sponding to the splitting on whichπ1(S)acts.

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We shall be interested in the universal cover S Gel of S Gel.Paths in S Gel and S Gel will be called electric paths(following Farb[12]).Geodesics and quasigeodesics in the electric metric will be called electric geodesics and electric quasigeodesics respectively.

De?nitions:

?A pathγ:I→Y in a path metric space Y is a K-quasigeodesic if we have

L(β)≤KL(A)+K

for any subsegmentβ=γ|[a,b]and any recti?able path A:[a,b]→Y with the same endpoints.

?γis said to be an electric K,?-quasigeodesic in S Gel without backtrack-ing ifγis an electric K-quasigeodesic in S Gel andγdoes not return to any any lift S A? S Gel(of an amalgamation component S A?S)after leaving it.

We collect together certain facts about the electric metric that Farb proves in[12].N R(Z)will denote the R-neighborhood about the subset Z in the hyperbolic metric.N e R(Z)will denote the R-neighborhood about the subset Z in the electric metric.

Lemma3.1(Lemma4.5and Proposition4.6of[12])

1.Electric quasi-geodesics electrically track hyperbolic geodesics:Given

P>0,there exists K>0with the following property:For some S Gel, letβbe any electric P-quasigeodesic without backtracking from x to y, and letγbe the hyperbolic geodesic from x to y.Thenβ?N e K(γ).

2.Hyperbolicity:There existsδsuch that each S Gel isδ-hyperbolic,inde-

pendent of the curveσwhose lifts are electrocuted.

Note:As pointed out before,S Gel is quasi-isometric to a tree and is therefore hyperbolic.The above assertion holds in far greater generality than stated.We discuss this below.

We consider a hyperbolic metric space X and a collection H of(uni-formly)C-quasiconvex uniformly separated subsets,i.e.there exists D>0 such that for H1,H2∈H,d X(H1,H2)≥D.In this situation X is hyper-bolic relative to the collection H.The result in this form is due to Klarreich [19].We give the general version of Farb’s theorem below and refer to[12] and Klarreich[19]for proofs.

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Lemma3.2(See Lemma4.5and Proposition4.6of[12]and Theorem5.3 of Klarreich[19])Givenδ,C,D there exists?such that if X is aδ-hyperbolic metric space with a collection H of C-quasiconvex D-separated sets.then,

1.Electric quasi-geodesics electrically track hyperbolic geodesics:Given

P>0,there exists K>0with the following property:Letβbe any electric P-quasigeodesic from x to y,and letγbe the hyperbolic geodesic from x to y.Thenβ?N e K(γ).

2.γlies in a hyperbolic K-neighborhood of N0(β),where N0(β)denotes

the zero neighborhood ofβin the electric metric.

3.Hyperbolicity:X is?-hyperbolic.

A special kind of geodesic without backtracking will be necessary for uni-versal covers S Gel of surfaces with some electric metric.Letσ,Aσbe as before.

Letλe be an electric geodesic in some( S Gel,d Gel).Then,each segment ofλe between two lifts Aσof Aσ(i.e.lying inside a lift of an amalgamation component)is required to be perpendicular to the bounding geodesics.We shall refer to these segments ofλe as amalgamation segments because they lie inside lifts of the amalgamation components.

Let a,b be the points at whichλe enters and leaves a lift Aσof Aσ.If a,b lie on the same side,i.e.on a lift of eitherσ×{0}orσ×{1},then we join a,b by the geodesic joining them.If they lie on opposite sides of Aσ, then assume,for convenience,that a lies on a lift ofσ×{0}and b lies on a lift ofσ×{1}.Then we join a to b by a union of2geodesic segments [a,c]and[d,b]lying along σ×{0}and σ×{1}respectively(for some lift Aσ),along with a‘horizontal’segment[c,d],where[c,d]? Aσprojects to a segment of the form{x}×[0,1]?σ×[0,1].We further require that the sum of the lengths d(a,c)and d(d,b)is the minimum possible.The union of the three segments[a,c],[c,d],[d,b]shall be denoted by[a,b]int and shall be referred to as an interpolating segment.See?gure below.

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a b

c d

Figure2:Interpolating segment

Remark:We note?rst that if we collapse each lift of Aσalong the

I(=[0,1])-?bres,(and thus obtain a geodesic that is a lift ofσ),thenλGel

becomes an electric geodesicλel in the universal cover S el of S el.Here S el denotes the space obtained by electrocuting the geodesicσ(See Section3.1

of[30].

Let c:S G→S be the map that collapses I-?bres,i.e.it maps the

annulus Aσ=σ×I to the geodesicσby taking(x,t)to x.The lift?c: S G→ S

17

collapses each lift of Aσalong the I(=[0,1])-?bres to a geodesic that is a lift ofσ).Also it takesλGel to an electric geodesicλel in the universal cover S el of S el(thatλel is an electric geodesic in S el follows easily,say from normal forms).These were precisely the electro-ambient quasigeodesics in the space S e l(See Section3.1of[30]for de?nitions).

Remark:The electro-ambient geodesics in the sense of[30]and those in the present paper di?er slightly.The di?erence is due to the grafting annulus Aσthat we use here in place ofσ.What is interesting is that whether we electrocuteσ(to obtain S el)or its complementary components (to obtain S Gel),we obtain very nearly the same electro-ambient geodesics. In fact modulo c,they are the same.

We now recall a Lemma from[30]:

Lemma3.3(See Lemma3.7of[30])There exists(K,?)such that each electro-ambient representativeλel of an electric geodesic in S el is a(K,?) hyperbolic quasigeodesic.

Since?c is clearly a quasi-isometry,it follows easily that:

Lemma3.4There exists(K,?)such that each electro-ambient representa-tiveλGel of an electric geodesic in S Gel is a(K,?)hyperbolic quasigeodesic.

In the above form,electro-ambient quasigeodesics are considered only in the context of surfaces,closed geodesics on them and their complementary (amalgamation)components.A considerable generalisation of this was ob-tained in[30],which will be necessary while considering the global geometry of M(rather than the geometry of B,for an amalgamated building block B).

We recall a de?nition from[30]:

De?nitions:Given a collection H of C-quasiconvex,D-separated sets and a number?we shall say that a geodesic(resp.quasigeodesic)γis a geodesic (resp.quasigeodesic)without backtracking with respect to?neighbor-hoods ifγdoes not return to N?(H)after leaving it,for any H∈H.A geodesic(resp.quasigeodesic)γis a geodesic(resp.quasigeodesic)without backtracking if it is a geodesic(resp.quasigeodesic)without backtracking with respect to?neighborhoods for some?≥0.

Note:For strictly convex sets,?=0su?ces,whereas for convex sets any?>0is enough.

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Let X be aδ-hyperbolic metric space,and H a family of C-quasiconvex, D-separated,collection of subsets.Then by Lemma3.2,X el obtained by electrocuting the subsets in H is a?=?(δ,C,D)-hyperbolic metric space. Now,letα=[a,b]be a hyperbolic geodesic in X andβbe an electric P-quasigeodesic without backtracking joining a,b.Replace each maximal subsegment,(with end-points p,q,say)starting from the left ofβlying within some H∈H by a hyperbolic geodesic[p,q].The resulting connected pathβq is called an electro-ambient representative in X.

In[30]we noted thatβq need not be a hyperbolic quasigeodesic.However, we did adapt Proposition4.3of Klarreich[19]to obtain the following:

Lemma3.5(See Proposition4.3of[19],also see Lemma3.10of[30]) Givenδ,C,D,P there exists C3such that the following holds:

Let(X,d)be aδ-hyperbolic metric space and H a family of C-quasiconvex, D-separated collection of quasiconvex subsets.Let(X,d e)denote the electric space obtained by electrocuting elements of H.Then,ifα,βq denote respec-tively a hyperbolic geodesic and an electro-ambient P-quasigeodesic with the same end-points,thenαlies in a(hyperbolic)C3neighborhood ofβq.

Note:The above Lemma will be needed while considering geodesics in M.

3.2Electric isometries

Recall that S G is a grafted surface obtained from a(?xed)hyperbolic metric by grafting an annulus Aσin place of a geodesicσ.

Now letφbe any di?eomorphism of S G that?xes Aσpointwise and(in case(S G?Aσ)has two components)preserves each amalgamation compo-nent as a set,i.e.φsends each amalgamation component to itself.Such aφwill be called a component preserving di?eomorphism.Then in the electrocuted surface S Gel,any electric geodesic has length equal to the number of times it crosses Aσ.It follows thatφis an isometry of S Gel.(See Lemma3.12of[30]for an analogous result in S el.)We state this below. Lemma3.6Letφdenote a component preserving di?eomorphism of S G. Thenφinduces an isometry of(S Gel,d Gel).

Everything in the above can be lifted to the universal cover S Gel.We let φdenote the lift ofφto S Gel.This gives

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