Quasi-isometries preserve the geometricdecomposition of Haken manifolds�Michael Kapovich and Bernhard Leeb��December 29, 1995Abstract. We prove quasi-isometry invariance of the canonical decomposition forfundamental groups of Haken 3-manifolds with zero Euler characteristic. We showthat groups quasi-isometric to Haken manifold groups with nontrivial canonical de-composition are �nite extensions of Haken orbifold groups. As a by-product wedescribe all 2-dimensional quasi- ats in the universal covers of non-geometric Hakenmanifolds. 1Contents1 Introduction 22 Preliminaries 42.1 Notations and conventions . . . . . . . . . . . . . . . . . . . . . . . . 42.2 3-manifolds and their canonical decomposition . . . . . . . . . . . . . 52.3 Ultralimits and asymptotic cones . . . . . . . . . . . . . . . . . . . . 62.4 Busemann functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.5 Quasi-isometric embeddings into piecewise Euclidean spaces . . . . . 73 Asymptotic cones of universal covers of Haken manifolds 83.1 Geometric components . . . . . . . . . . . . . . . . . . . . . . . . . . 83.2 Separation by ats . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93.3 Projections to ats . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113.4 Rigidity of bilipschitz homeomorphisms . . . . . . . . . . . . . . . . . 113.5 Structure of bilipschitz-embedded ats . . . . . . . . . . . . . . . . . 134 Quasi-isometries of universal covers of Haken manifolds 144.1 Linear divergence of quasi-disks . . . . . . . . . . . . . . . . . . . . . 154.2 Rigidity of quasi-isometries . . . . . . . . . . . . . . . . . . . . . . . . 164.3 Structure of quasi- ats . . . . . . . . . . . . . . . . . . . . . . . . . . 17�To appear in Inventiones Mathematicae��This research was partially supported by the SFB 256 in Bonn and the NSF grant DMS-9306140(Kapovich).11991 Mathematics Subject Classi�cation: 20F32, 53C21, 57M50.1

5 Groups quasi-isometric to fundamental groups of Haken manifolds 195.1 Quasi-actions of groups on metric spaces . . . . . . . . . . . . . . . . 195.2 Quasi-actions on geometric components . . . . . . . . . . . . . . . . . 205.3 The general case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21Bibliography 221 IntroductionLet � be a �nitely generated group. A geodesic metric space X, on which � acts prop-erly discontinuously and cocompactly by isometries, can be regarded as a geometricmodel for �. Important examples are Cayley graphs associated to �nite generat-ing sets and universal covers of compact Riemannian manifolds with fundamentalgroup �. All such model spaces X are quasi-isometric to one another and theirquasi-isometry invariants are called geometric invariants of �, cf. [Gr1]. It is a basicquestion in this context to classify all �nitely generated groups up to quasi-isometry.Note that commensurable groups have the same geometric invariants, whereas theconverse is in general not true.This paper deals with the geometry of 3-manifold groups. Our main result con-cerns the canonical decomposition of Haken manifolds M with boundary of zero Eu-ler characteristic. Jaco, Shalen, Johannson and Thurston proved that M can be cutalong at surfaces into �nitely many geometric components which are either Seifertor hyperbolic. This canonical decomposition of M is unique up to isotopy and itcorresponds to an algebraic decomposition of �1(M) as a graph of groups which isinvariant under group automorphisms. We prove that the canonical decompositionis more generally invariant under all quasi-isometries and therefore it is a geometricinvariant of the fundamental group. To make this precise, put a Riemannian metricon M and take the universal cover X = ~M as a geometric model for �1(M). Thecanonical decomposition of M lifts to a decomposition of X where a geometric com-ponent of X is the universal cover of a geometric component of M . Let X 0 = ~M 0 bethe universal cover of another Haken manifold M 0 of the same kind, decomposed inthe same way.Main Theorem 1.1 Let � : X ! X 0 be a quasi-isometry. Then � preserves thegeometric decompositions of X and X 0 in the following sense: For any geometric com-ponent Y of X there exists a geometric component Y 0 of X 0 within uniformly boundedHausdor�-distance from �(Y ). The components Y and Y 0 have the same type (Seifertor hyperbolic). Moreover � preserves the ordering of geometric components and there-fore induces an isomorphism of the trees dual to the geometric decompositions of Xand X 0.We did a �rst step in this direction in our earlier paper [KL2] where we provedthat the quasi-isometry class of �1(M) detects whether M has a Seifert component.Theorem 1.1 implies that also the existence of a hyperbolic component is \visible"in the geometry of �1(M). This had �rst been proven by N. Brady and Gerstenusing di�erent techniques; they showed that the divergence of �1(M) is exponential2

if and only if M has a hyperbolic component, see [Ge]. Note that there are non-geometric Haken manifolds whose fundamental groups are quasi-isometric but notcommensurable, see [KL1, KL3].Our main application is a geometric characterization of Haken manifold groups:Theorem 1.2 Suppose that � is a �nitely generated group whose Cayley graphs arequasi-isometric to the universal cover X of a non-geometric Haken manifold M withzero Euler characteristic. Then there is a short exact sequence1 �! �nite group �! � �! �1(O) �! 1where O is a compact 3-dimensional orbifold which is �nitely covered by a Hakenmanifold of the same kind asM . In particular, if � is torsion-free then � is isomorphicto the fundamental group of a Haken manifold N .Results analogous to Theorem 1.2 were previously known in several cases whenM is geometric: The result for Nil-manifolds is due to Gromov, see [Gr2], and theeuclidean case is due to Bridson and Gersten. Rie�el [R] proved Theorem 1.2 whenMis a Seifert manifold with hyperbolic base orbifold. The case whenM is a Sol-manifoldremains open. Note that there are obvious examples of self quasi-isometries of Solwhich are not within bounded distance from any isometry. Regarding the hyperboliccase, Schwartz [Sch] proved for �nite volume noncompact hyperbolic manifolds H ofdimension � 3 that the quasi-isometry group of �1(H) is naturally isomorphic to thecommensurator of �1(H); this shows in particular that any �nitely generated groupquasi-isometric to �1(H) is a �nite extension of a group commensurable with �1(H).As a by-product of the proof of Theorem 1.1 we also give a classi�cation of 2-dimensional quasi- ats in X, cf. Theorem 4.10. We prove that each 2-quasi- at inX is contained in a tubular neighborhood of a �nite union of isolated ats in X.Besides quasi- ats which are Hausdor� close to a at there are also twisted quasi- atswhich spread through a �nite chain of consecutive Seifert components. We describecanonical models for these quasi- ats in Section 4.3.Our approach is based on the strong link between Haken 3-manifolds and thegeometry of nonpositive curvature. Based on Thurston's Hyperbolization Theorem, itis shown in [L] that Haken manifoldsM of zero Euler characteristic generically admitmetrics of nonpositive curvature with totally geodesic at boundary. Moreover, weprove in [KL3] that their fundamental groups are geometrizable in the following weaksense: If M is neither a Sol- nor Nil-manifold, then there exists a Haken manifoldof nonpositive curvature M 0 such that the universal covers X = ~M and X 0 = ~M 0are bilipschitz homeomorphic by a homeomorphism which preserves the geometricdecomposition. In particular, the geometric components of the universal cover X arequasi-isometrically embedded and X is bicombable. Therefore in the present paperwe shall only consider nonpositively curved Haken manifolds M .We already mentioned that single 2-quasi- ats are generally not Hausdor�-closeto ats. Our idea is to show that su�ciently complicated patterns of 2- ats are quasi-isometrically rigid. As in [KL2], we use the concept of asymptotic cone of a metricspace. arguments A quasi-isometry X ! X 0 becomes \more continuous" when onerescales the metrics with a small factor, and in the ultralimit it induces a bilips-chitz homeomorphism X! ! X 0! of asymptotic cones. By analysing the geometry3

and topology of the asymptotic cones we prove that their induced geometric decom-positions are preserved by bilipschitz homeomorphisms. This is done by classifyingbilipschitz embedded 2- ats and topologically characterizing the ats separating geo-metric components. The Divergence Lemma 4.1 is the key tool for translating thetopological rigidity of the asymptotic cones into a statement about quasi-isometricrigidity for the geometric decompositions of X and X 0. This lemma implies that a 2-dimensional quasi- at Q in X which is not uniformly close to a convex set C divergesfrom C at a linear rate and hence the distinction between Q and C becomes visiblein the asymptotic cone X!.Regarding Theorem 1.2, a �nitely generated group, which is quasi-isometric to X,admits an action by uniform quasi-isometries on X. The Main Theorem implies thatthe canonical decomposition of X is quasi-preserved and there is an induced actionon the dual tree for the canonical decomposition. By a general argument, the vertexand edge stabilizers act quasi-transitively on the corresponding components and ats.Using work of Schwartz and Tukia we conclude that the vertex stabilizers are �niteextensions of fundamental groups of 3-dimensional hyperbolic and Seifert orbifoldsand the edge stabilizers are their peripheral subgroups.2 Preliminaries2.1 Notations and conventionsWe will use di�erent notions of distance between subsets A;B of metric spaces: theHausdor� distance and the distance of closest points denoted by dist(A;B). Thedistance of a point x from a set A will be sometimes denoted by dA(x). If A is asubset of a topological space X then �A will denote the closure of A in X. We use thenotation [xy] for the geodesic segment connecting points x; y in a metric space X. IfZ is a subset in a metric space X and R > 0 then NR(Z) denotes the R-neighborhoodof Z in X, we will refer to NR(Z) as a tubular neighborhood of Z. We assume thatall segments, rays and geodesics are parameterized by unit speed.A map f : (X1; d1)! (X2; d2) of metric spaces is a (K; �)-quasi-isometric embed-ding with K; � > 0 ifK�1d1(x; y)� � � d2(f(x); f(y)) � Kd1(x; y) + �for each x; y 2 X1. (One should think of � as a large positive number.) Note thatquasi-isometric embeddings are not necessarily injective or continuous. A map f1 :(X1; d1)! (X2; d2) is a quasi-isometry if there are two constants C1; C2 and anothermap f2 : (X2; d2) ! (X1; d1) such that both f1; f2 are quasi-isometric embeddingsand d1(f2f1(x); x) � C1; d2(f1f2(y); y) � C2for every x 2 X1; y 2 X2. Such spaces X1; X2 are called quasi-isometric.Hadamard (or CAT (0)) spaces are complete geodesic metric spaces with nonposi-tive curvature in the distance comparison sense, cf. [Ba, KlL]. They are not assumedto be locally compact. In Hadamard spaces one can de�ne the angle between geodesicsegments [ab]; [ac], see [KL2]. We shall denote by @geoX the geometric or ideal boun-dary of the Hadamard space X. For a closed convex subset C in a Hadamard space X,4

projC will denote the closest-point projection to C. These projections are distance-nonincreasing. A at, bilipschitz at, respectively quasi- at in X is (the image of) anisometric, bilipschitz, respectively quasi-isometric embedding of the euclidean 2-planeinto X.Convention 2.1 All ats, bilipschitz ats and quasi- ats considered in the presentpaper are 2-dimensional.2.2 3-manifolds and their canonical decompositionWe refer to [S] for information about the geometrization of 3-manifolds and a descrip-tion of the eight 3-dimensional homogeneous geometries. Here we only recall a fewfacts which are important for this paper.A compact smooth 3-manifold P is called geometric if its interior admits a geo-metric structure, i.e. a complete locally homogeneous Riemannian metric. If P isaspherical and has nonempty boundary, then the occurring homogeneous spaces willbe H 3 and H 2 � R. In the latter case, the manifold P is Seifert and itself admits ametric modelled on H 2�R with totally geodesic boundary, unless it has almost abelianfundamental group. Manifolds P locally modelled on H 3 -geometry are called hyper-bolic. Note that according to our de�nition the solid torus D2�S1 and S1�S1� [0; 1]are geometric manifolds which are hyperbolic and Seifert simultaneously.In this paper we will only consider aspherical 3-manifolds of zero Euler charac-teristic, equivalently the boundary of such manifolds is a (possibly empty) collectionof tori and Klein bottles. Instead of giving the de�nition of Haken manifolds, weremind that all non-geometric Haken manifolds of zero Euler characteristic admit acanonical decomposition into �nitely many geometric manifolds which are glued alongboundary tori or Klein bottles, see [JS, J, Th, Ka, O]. This decomposition is uniqueup to isotopy if these geometric submanifolds are chosen to be maximal up to isotopy.Note that geometric components of a Haken manifold of this kind never have almostabelian fundamental group, thus the classes of hyperbolic and Seifert componentsbecome disjoint.If the Haken manifoldM of zero Euler characteristic carries a Riemannian metricg of nonpositive sectional curvature with totally-geodesic at boundary, then thecanonical topological decomposition of M into hyperbolic and Seifert componentscan be realized geometrically by cutting along totally-geodesically embedded at toriand Klein bottles �, cf. [L, LS]. The metric g can be chosen (once di�erentiable) sothat all Seifert components are in their interior locally isometric to H 2�R. Call a atin the universal cover X ofM an isolated at if it either covers one of the at surfaces� in the canonical decomposition of M or is a boundary at of X. This terminologyis motivated by the fact that a at is parallel to an isolated at if and only if no other at intersects it transversally. The isolated ats decompose X into convex subsetswith totally-geodesic at boundary which we call geometric components of X; theyare universal covers of the geometric components of M . Each Seifert component Yof X splits isometrically as the product of the real line and a 2-dimensional factorwhich is isometric to a convex domain in H 2 bounded by disjoint geodesics. The�bration of Y by parallel geodesics covers a Seifert �bration of the correspondingSeifert component of M by closed geodesics.5

2.3 Ultralimits and asymptotic conesFor a discussion of ultralimits of metric spaces we refer to [DW, Gr2, KL2, KlL], herewe recall some of their basic properties.Fix a non-principal ultra�lter ! on N. For any sequence of pointed metric spaces(Xn; x0n) the ultralimit (X!; x0!) is a metric space. Points in the ultralimit are rep-resented by sequences (xn) of points in Xn. If the sequence (Xn) is precompact inthe Gromov-Hausdor� topology, then the ultralimit is the Gromov-Hausdor� limit ofa !-large subsequence. Ultralimits of Hadamard spaces are Hadamard spaces. Theultralimit of a sequence of (Kn; �n)-quasi-isometries fn : (Xn; x0n) ! (Yn; y0n) is a(K; �)-quasi-isometry f! : (X!; x0!)! (Y!; y0!) with K = !- limKn and � = !- lim �n.To form the asymptotic cone X! of a metric space X, one chooses sequencesof scale factors �n > 0 with !- lim�n = 0 and basepoints x0n 2 X and takes theultralimit (X!; x0!) := !- limn(�n � X; x0n). Here, �n � X denotes the metric spaceobtained by rescaling the metric of X with the factor �n. When we speak of \theasymptotic cone X! of X", we mean one of these ultralimits, suppressing the choicesof �n; x0n in our notation. In general, the isometry type ofX! depends on these choices.However, in our applications various asymptotic cones will share the same geometricproperties. The asymptotic cone X! of a Hadamard space X is a Hadamard spaceand in general not locally compact. If X admits a cocompact group of isometriesthen X! is homogeneous.The asymptotic cone is a useful tool for the study of quasi-isometries, because a(K; �)-quasi-isometric embedding of metric spaces becomes continuous in the rescalingprocess and induces a K-bilipschitz embedding of their asymptotic cones.2.4 Busemann functionsSuppose that Y is a Hadamard space. Busemann functions measure the relativedistance from points at in�nity. Pick a point � in the geometric boundary of Y ,i.e. an equivalence class of parallel geodesic rays. Take a unit speed geodesic ray� : [0;1) ! X asymptotic to �. The Busemann function B� corresponding to � isthe monotonic limit : B�(�) := limt!1[d(�; �(t))� t]B� is a convex function which is well-de�ned up to an additive constant. For everyray r asymptotic to � we have:B� � r(t) = �t + const (1)If r1; r2 are two geodesic rays asymptotic to �, then B� � r1(t) = B� � r2(t) holds i�the ideal triangle with vertices r1(t); r2(t); � has angles � �=2 for all t. These twoproperties characterize Busemann functions.Level sets of Busemann functions are called horospheres. The sublevel sets arecalled horoballs and they are convex.We prove two facts about Busemann functions for later reference.Lemma 2.2 Suppose that C is a convex subset of Y such that the Busemann functionB� is constant on C. Then the union U of geodesic rays emanating from points of Cand asymptotic to � is convex and splits isometrically as the direct productC � R+ :6

Proof: For a point q 2 C let lq denote the geodesic ray emanating from q in thedirection �. Let x; y be a pair of distinct points in C, the function B� is constant on[xy]. It follows from the de�nition of the Busemann function that the angles betweenlx; ly and the geodesic segment [xy] are at least �=2. However Y is a Hadamardspace, thus the ideal triangle with vertices x; y; � has angles �=2; �=2; 0 and spans anisometrically embedded at half-strip, namely the union[q2[xy]lq:The Lemma follows. 2Lemma 2.3 Let (Xn; x0n) be a sequence of based Hadamard spaces with the ultralimit(X!; x0!). Let Yn � Xn be convex subsets with !- limn d(x0n; Yn) = 1. Then f :=!- limn(dYn � dYn(x0n)) is a Busemann function on X!.Proof: For xn 2 Xn � Yn, denote by �xn : [0; dYn(xn)] ! Xn the perpendicular fromxn to Yn. For any points xn; x0n 2 Xn, the function t 7! d(�xn(t); �x0n(t)) decreasesmonotonically. If x! = (xn) is a point in X! then the ultralimit �x! := !- lim�xn :[0;1) ! X! is a geodesic ray which does not depend on the choice of the sequence(xn) representing x!, and all rays �x! are asymptotic to the same ideal point �! 2@geoX!. Applying the triangle inequality, we obtaind(�xn(t); x0n) � d(x0n; projYn(xn))| {z }�dYn (x0n) � dYn(�xn(t))| {z }=dYn (xn)�t � t + (dYn(x0n)� dYn(xn)):Passing to the ultralimit yieldsd(�x!(t); x0!) � t + (B(x0!)� B(x!))where B := !- limdYn. We rewrite the previous inequality:d(x0!; �x!(t))� t � B(x0!)� B(x!)and send t to in�nity: B�!(x0!)�B�!(x!) � B(x0!)�B(x!):Since we may exchange the roles of x! and x0!, the equality holds and we concludethat the function B�! �B is constant on X!. 2Remark 2.4 Similarly, one shows that the ultralimit of Busemann functions on Xnis a Busemann function on X! (or constant �1).2.5 Quasi-isometric embeddings into piecewise Euclidean spa-cesSuppose that X is a geodesic metric space and E � X is an open subset which isisometric to a convex subset in the Euclidean m-space Rm .7

Lemma 2.5 There is a constant R = R(K; �;m) such that the following is true: Letqf : Rm ! X be a (K; �)-quasi-isometric embedding with the image Q. Then eitherNR(Q) � E or Q \ E � NR(@E).Proof: If the conclusion of the lemma is not satis�ed then the convex subset E 0 :=E �N 2R3 (@E) satis�es Q\E 0 6= ; and E 0 6� NR3 (Q). Thus there exists a point x 2 Ewith d(x;Q) = R3 and d(x; @E) � 2R3 :If the assertion of the lemma were not true, then we can �nd sequences of subsetsEn � Xn, (K; �)-quasi-isometries qfn : Rm ! Xn, points xn 2 En and numbers Rntending to in�nity, so that: d(xn; qfn(Rm)) = d(xn; qfn(0)) = Rn3 and d(xn; @En) �2Rn3 . Now we rescale Rm with the center 0 and Xn with the center xn using thescale factor R�1n . The ultralimit of the (K;R�1n �)-quasi-isometric embeddings qfn :(R�1n �Rm ; 0)! (R�1n �Xn; xn) is a K-bilipschitz embedding [ : Rm ! X!. The subsetE! := !- limEn � X! is isometric to a convex subset of Rm . Moreover its interior(with respect to an isometric embedding into Rm) is open in X!, because the Xnwere assumed to be geodesic. By construction, the image of [ intersects the interiorof E!, but does not contain it. This is impossible, because the restriction of [ to[�1(int(E!)) is a bilipschitz map between open subsets of Rm and therefore a localhomeomorphism. 2Corollary 2.6 (H. F�urstenberg) Every (K; �)-quasi-isometric embedding Rm ! Rmis a (K; �0)-quasi-isometry with a constant �0 = �0(K; �;m).3 Asymptotic cones of universal covers of HakenmanifoldsLet Hnpc be the class of all compact non-geometric Haken manifolds which areequipped with a nonpositively curved Riemannian metric such that the boundary istotally-geodesic and at. These manifolds have geometric rank one and they containtotally-geodesically immersed at 2-tori. Throughout this section, M (resp. M 0) shalldenote a Riemannian manifold in Hnpc and X (resp. X 0) its universal cover. The de-composition of X into geometric components (cf. section 2.2) induces a correspondingdecomposition of X! and we will prove that this decomposition is preserved by bilip-schitz homeomorphisms X! ! X 0!. A complete description of bilipschitz embedded ats will be given in section 3.5; besides ats there are non-trivial twisted examplesof bilipschitz ats. Please keep in mind convention 2.1: all ats and bilipschitz atsare assumed to be 2-dimensional!3.1 Geometric componentsLet Yn � X, n 2 N, be geometric components with !- lim��1n �d(Yn; x0n) <1. We calltheir ultralimit Y! = !- lim��1n � Yn � X! a geometric component in the asymptoticcone X!. Since there are only �nitely many isometry types of geometric componentsin X, Y! is isometric to the asymptotic cone of a geometric component Y of X. We8

call Y! Seifert, respectively hyperbolic, according to the type of Y . We collect someproperties of the geometric components proven in [KL2].A sequence of ats Fn � Yn with !- lim��1n � d(Fn; x0n) < 1 determines a atF! � Y!. In fact, every at in Y! arises in this way. If Y! is hyperbolic, then the atsFn can be chosen in @Yn. Moreover, by Lemma 2.14 and Corollary 4.6 in [KL2], eachbilipschitz embedded at in Y! is totally-geodesic:Lemma 3.1 Any bilipschitz at in X! which is contained in a geometric componentis a at.Seifert components are isometric to the product of R with a metric tree whichbranches everywhere and is geodesically complete. A key property of hyperboliccomponents is that any two distinct ats share at most one point, cf. Proposition4.3 in [KL2]. It follows that Seifert components cannot be embedded into hyperboliccomponents:Lemma 3.2 Let T be a metric tree with at least 3 ideal end points. Then T � Rcannot be bilipschitz embedded into a hyperbolic component Y!.Later we will need the following lemma concerning the separation of Seifert com-ponents by lines:Lemma 3.3 Let T be a geodesically complete tree and C � R a closed subset. Assumethat f : C ! T � R is a bilipschitz embedding whose image l separates. Then C =R and projT (l) is contained in a segment with no branch point in its interior. Inparticular, if T branches everywhere then l is a �ber ftg � R.Proof: Suppose that projT (l) contains two points a; b which form a tripod togetherwith a third point c. If F � T � R is a at which doesn't contain a and b, then F \ lis a proper subset of l and consequently doesn't separate F (Alexander duality). Forany points x; y 2 T � R there are ats Fx; Fy containing c so that x 2 Fx and y 2 Fy.Therefore x and y can be connected in (Fx[Fy)� l, a contradiction. Hence, projT (l)is contained in a segment with no branch point in its interior. If C 6= R, then f(C)cannot separate any at in T � R and we again obtain a contradiction. 23.2 Separation by atsSuppose that F is an isolated at adjacent to a geometric component Y � X. Werefer to the pair (F; Y ) as a cooriented at. For each cooriented isolated at (F; Y ),we introduce a partition LF of F into disjoint subsets: if Y is a Seifert component,then LF consists of the parallel lines corresponding to Seifert �bers; if Y is hyperbolic,then LF just consists of the points in F . A cooriented at (F; Y ) de�nes a signeddistance function sdF on X: we set sdF (x) := �dist(x; F ) with the positive sign ifx belongs to the same connected component of X � F as Y and the negative signotherwise.There are corresponding notions in the asymptotic cone. Namely, let (Fn; Yn) bea sequence of cooriented ats with signed distance functions sdFn. It gives rise toa cooriented at (F!; Y!) := !- lim(Fn; Yn) and a signed distance function sdF! :=9

!- lim sdFn on X!. We call the set fsdF! > 0g the positive side of F!. Note that thepositive side is not connected at all! Y! := (Yn) is the geometric component adjacentto F! on the positive side. If the negative side of F! is non-empty (equivalently, ifFn is not parallel to a boundary at of X for !-all n), then there is a geometriccomponent Z! of X! adjacent to F! on the negative side as well. In this situation wesay that the negative side of F! is the F!-side of Y!. Similarly, the positive side ofF! is the F!-side of Z!.The at F! inherits a partition LF! into points or parallel lines. If the geometriccomponent on the positive side is Seifert, then LF! consists of lines parallel to theR-�ber, otherwise it consists of points. If F! has adjacent Seifert components on bothsides, then the two families of lines which correspond to both coorientations of F! aretransversal.We say that a at or a geometric component F! of X! essentially separates twosets S1; S2 � X! if the sets Si � F! lie on distinct sides of F! ( we allow Si � F!).A set S � X! is said to be essentially split by F! if there are points of S � F! onboth sides of F!. There is a dual tree to the decomposition of X into geometriccomponents and this tree-like order persists in the asymptotic cone: For any threegeometric components Y1!; Y2!; Y3! of X! either one of them essentially separates theother two or there is a unique geometric component Y! which essentially separatesany two of the components Yi!. Observe also that the set of isolated ats whichessentially separate two given isolated ats F! and F 0! is totally ordered. We callisolated ats F1!; : : : ; Fn! inX! consecutive if Fi! and Fi+1;! belong to one componentand Fi! essentially separates the ats Fi�1;! for all i (where it makes sense). Notethat three consecutive isolated ats share at most one point. The same is true forfour consecutive geometric components.Lemma 3.4 Assume that the subset A � X! is not essentially split by any isolated at. Then A is contained in a single geometric component of X!.Proof: Let us �rst consider the case that A consists of two points x! and y!. Thereare isolated ats Fx! � x! and Fy! � y!. We are done if Fx! and Fy! coincide.Otherwise the set C of geometric components which essentially separate Fx! and Fy!is non-empty and totally ordered. By assumption, any component in C containsexactly one of the points x! and y!, because otherwise we are again done. At most�nitely many components in C can contain interior points of the segment [x!y!],since 4 consecutive components share at most one point. Hence there exist adjacentcomponents Y!; Y 0! 2 C each of which contains exactly one of the points x!; y!. Thenthe isolated at Y! \ Y 0! essentially separates x! and y!, a contradiction. Thus ourclaim holds if A consists of two points.Consider now the general case. Let a; b be any two points of A. As shown above,they lie in some geometric component Y0. If A is not already contained in Y0, thereis an isolated at F1 � Y0 which separates Y0 � F1 from a point c 2 A. Since F1does not split A, all of A � F1 lies on the same side of F1. In particular a; b belongto the geometric component Y1 6= Y0 adjacent to F1. If A is not contained in Y1,we can continue this argument inductively and construct four consecutive geometriccomponents Y0; Y1; Y2; Y3 which contain the points a; b. However, the intersection offour consecutive geometric components contains at most one point. 210

3.3 Projections to atsBasic to our understanding of the topology ofX! is the study of projections to isolated ats. Inside geometric components, we have:Lemma 3.5 Let F! be an isolated at in the geometric component Y! and let � � Y!be a geodesic segment disjoint from F!. Then projF!(�) is contained in a set l 2 LF!.Proof: The assertion for hyperbolic components is included in Lemma 4.4 of [KL2].The Seifert case follows from the following corresponding statement for trees:Sublemma 3.6 Let c be a geodesic in a metric tree and [uv] a geodesic segmentdisjoint from c. Then the nearest-point-projection to c maps [uv] to a point.Proof: Let p 2 c be the point closest to [uv] and q 2 [uv] be the point closest to c.Recall that if [rs] and [st] are geodesic segments in a tree with [rs] \ [st] = fsg then[rs] [ [st] = [rt]. This implies that any segment from a point on � to a point on ccontains [pq] and the claim follows. 2We extend the previous lemma to projections of the entire asymptotic cone X!.Proposition 3.7 Let F! be a cooriented isolated at in X! and suppose that A is aconnected component of X! � F! on the positive side of F!. Then projF!(A) � l forsome l 2 LF! and hence �A \ F! � l.Proof: Let x!; z! 2 X!�F! be points on the positive side of F! so that [x!z!]\F! = ;.Let Y! be the geometric component of X! adjacent to F! on the positive side.Case 1: If [x!z!] \ Y! = ; then there exists a unique isolated at F 0! � Y! suchthat F 0! essentially separates [x!z!] from Y!. Hence projF! [x!z!] � projF!F 0! whichis contained in a set l 2 LF! by the previous lemma.Case 2: If [x!z!]\Y! = [x0!z0!] then the previous lemma implies that projF! [x0!z0!]� l 2 LF! and by the same reasoning as in the Case 1 one has projF! [x!x0!] � l andprojF! [z!z0!] � l.Hence each geodesic segment on the positive side of F! and disjoint from F!projects into a set l 2 LF! . We conclude that the sets fsdF! > 0g \ proj�1F! (l) areopen for all l 2 LF! and our claim follows. 23.4 Rigidity of bilipschitz homeomorphismsWe �rst look at the position of a bilipschitz embedded at B = f(R2) in X! relativeto a cooriented isolated at F!. Suppose that B � F! consists of several connectedcomponents and let B0 be a component on the positive side of F!. By Proposition3.7, @B0 is contained in some l 2 LF! and therefore is homeomorphic to a closedsubset of the real line. Since @B0 separates B, Alexander duality yields that l = @B0is a line and the geometric component adjacent to F! on the positive side is Seifert.Note that the pair (B0; @B0) is homeomorphic to the pair (R+ � R; 0 � R).Lemma 3.8 Any at F! in X! is contained in a single geometric component. More-over, F! arises as the ultralimit of a sequence of ats in X.11

Proof: If F! is not contained in a geometric component then it is essentially split bysome isolated at F 0! (Lemma 3.4). The geometric components on the both sides ofF 0! must be Seifert. Moreover, F! \ F 0! contains two transversal lines and thereforeF! = F 0!. Since any at in a geometric component arises as the ultralimit of asequence of ats, the claim follows. 2Next we give a topological characterization of isolated ats which are not adjacentto Seifert components of X!.Lemma 3.9 Let B be a bilipschitz at in X!. The following two properties areequivalent:1. The intersection of B with any other bilipschitz at B0 contains at most onepoint.2. B is an isolated at which is not adjacent to any Seifert component.Proof: If B is a bilipschitz at which satis�es the �rst property then B cannot beessentially split by any isolated at. By Lemmata 3.4 and 3.1, B is a at contained in ageometric component. The component must be hyperbolic, so B is an isolated at andmoreover the geometric components on the both sides of B must be hyperbolic. (Notethat it may happen that B has only one side!)Vice versa, assume now that F is an isolated at satisfying the second propertyand let B0 be a bilipschitz at intersecting F . Then for any connected componentB0 of B0 � F , �B0 \ F is a point in F . Since B cannot be disconnected by one point,B0 � F consists of one component and B \ F is a point. 2Lemma 3.10 Let T be a geodesically complete tree which branches at every point.Then for any bilipschitz embedding f : T � R ! X!, the image is contained in aSeifert component and the map f preserves the Seifert �bration.Proof: Suppose that an isolated at F! essentially splits f(T � R). Let � be con-nected components of f(T � R) � F! which lie on di�erent sides of F!. Then theirboundaries are transversal straight lines l� in F!. On the other hand, the inverseimages f�1(l�) separate T � R and hence they are parallel lines by Lemma 3.3. Thisis impossible, because f is bilipschitz.Hence f(T � R) is not essentially split by any isolated at and therefore lies ina Seifert component by Lemmata 3.4 and 3.2. The second assertion follows fromLemma 3.1. 2We apply the above observations to show that homeomorphisms of asymptoticcones are rigid in the sense that they preserve the decomposition into geometriccomponents.Proposition 3.11 Let X;X 0 2 Hnpc and let � : X! ! X 0! be a bilipschitz homeo-morphism. Then:(i) � maps ats to ats.(ii) Each isolated at which is not adjacent to a Seifert component is mapped via� to an isolated at of the same kind.(iii) The image of each Seifert component of X! is a Seifert component of X 0!.12

Proof: Assertion (ii) follows from Lemma 3.9 and assertion (iii) from Lemma 3.10.According to Lemma 3.8, any at in X! lies in a geometric component. Thus forisolated ats between hyperbolic components Assertion (i) follows again from Lemma3.9 and for ats contained in Seifert components from Lemma 3.10. 23.5 Structure of bilipschitz-embedded atsLet f : R2 ,! X! be a C-bilipschitz embedding. We will now take a closer look atthe position of the bilipschitz at B := f(R2) relative to an isolated at F whichseparates B, i.e. B � F is disconnected. We observed in the beginning of Section3.4 that, for each component B0 of B � F , projF (B0) is a straight line contained inB \ F . It follows that projF jF : F ! B \ Fis a retraction and B \ F is contractible.Assume that B1 and B2 are two components of B � F on the same side of F .Then li := �Bi \ F are parallel lines bounding a at strip S � F . Any points p1 2 l1and p2 2 l2 can be connected inside B \ F by a recti�able curve of length at mostC2 � d(p1; p2). Indeed, connect the points x1 = f�1(p1); x2 = f�1(p2) by the geodesicsegment [x1x2] � R2 , its length is at most Cd(p1; p2). Then the projection of f([x1x2])to F has length at most C2 � d(p1; p2) and lies inside of B \ F .Since B \ F is contractible we conclude that S � B \ F andB = B1 [ S [B2:Consider now the case that B1 and B2 are components of B � F on distinct sides ofF . Then li := �Bi \ F are transversal straight lines in F and, by the above,B = B1 [B2 [ (B \ F ):The set D := f�1(F � (l1[ l2)) = f�1proj�1(F � (l1[ l2)) is open in R2 , and thereforef jD : D ,! F � (l1 [ l2) is a local homeomorphism. On the other hand, f(R2) isclosed because f is proper, and f(D) must be a union of connected components ofF � (l1 [ l2). The lines l1; l2 divide F into four \quadrants" and we conclude thatB \ F is a union of two opposite closed quadrants.Now we are ready to discuss the structure of a bilipschitz at B which is notcontained in a single geometric component. We describe an inductive process ofgeometric decomposition of B. According to Lemma 3.4, B is essentially split by a at F0. Let B+ be the component of B � F0 on the positive side of F0. If B+ iscontained in the Seifert component S1 adjacent to F0 on the positive side then it isa vertical half-plane, as follows for instance from Lemma 3.1. In this case, we stopthe decomposition on the right side of F0. Otherwise, another isolated at F1 � S1essentially splits B. Between the pairs of quadrants f�1(F0\B) and f�1(F1\B) thereis a strip A1 whose image f(A1) is a at strip in S1. (This strip could degenerate toa single line.) We continue this process of decomposition on the both sides of F0 andobtain a sequence of consecutive Seifert components : : : ; S�1; S0; S1; : : :. The unionof these Seifert components is a convex subset of the asymptotic cone. The transitionof Q between adjacent Seifert components contributes a de�nite amount of stretch tothe bilipschitz embedding f , and this leads to:13

Lemma 3.12 The number of possible Seifert components Sj occurring in the decom-position is �nite and bounded uniformly in terms of the bilipschitz constant of f andthe geometry of M .Proof: Observe that �bres li : R ! Si and lj : R ! Sj in di�erent Seifert components,which are parameterized by unit speed, have uniform divergence. Namely there is apositive constant �, which depends on the angles between �bers of adjacent Seifertcomponents in M , so that limt!1 d(li(t); lj(t))t � �We denote the bilipschitz constant of f by C and restrict our attention to a �nitenumber N of Seifert components Sj. The points (1=t) � f�1 � lj(t) are contained in adisk of radius C + o(t) in the Euclidean plane and they are (�=C + o(t))-separated.Hence N is bounded in terms of �;C and the assertion follows. 2We summarize the above discussion.Description of bilipschitz 2- ats B in X!: Either B is contained in a geometriccomponent and is a genuine at. If this is not the case, we call B twisted. B isthen contained in a �nite collection of consecutive Seifert components S0; : : : ; Sk withk � 1. The consecutive isolated ats Fi := Si�1 \ Si are the isolated ats whichessentially split B. We describe the intersections as we move through the chain ofSeifert pieces: (B \ S0) � F1 and (B \ Sk)� Fk are vertical half-planes H0 and Hk.Let l+i and l�i be the lines in Fi which consists of points closest to Fi+1 and Fi�1.Furthermore, let l�1 and l+k be the boundaries of the half-planes H0 and Hk. Then theintersection B \ Fi is the union of two opposite quadrants bounded by l�i . Finally,the intersection B \Si, 0 < i < k, consists of the vertical strip Vi bounded by l+i ; l�i+1and four quadrants. The convex hull ch(B) of the bilipschitz at B is given by:ch(B) = H0 [ F1 [ V1 [ F2 [ : : : [ Fk [HkLemma 3.13 No bilipschitz at in X! is contained in a horoball.Proof: It follows from the description of bilipschitz ats in X! that if a convex setcontains a bilipschitz at then it also contains a at. Therefore, if a horoball containsa bilipschitz at, the corresponding Busemann function is bounded from above andhence is constant on a 2- at. Lemma 2.2 implies thatX! must contain a 3-dimensionalEuclidean half-space H. We know that any 2- at in X! is contained in a geometriccomponent. Since parallel 2- ats must be contained in the same geometric component,H itself lies in a geometric component, which is absurd. 24 Quasi-isometries of universal covers of HakenmanifoldsIn this section, X;X 0 will denote the universal covers of nonpositively curved Rie-mannian 3-manifolds M;M 0 2 Hnpc. 14

4.1 Linear divergence of quasi-disksWe want to understand the position of quasi- ats relative to convex subsets in X.The following local statement will be our basic tool. A quasi-disk is de�ned as (theimage of) a (K; �)-quasi-isometric embeddingqd : BR(0) � R2 ! Xof a Euclidean 2-disk for positive constants R;K and �.Divergence Lemma 4.1 There are positive functions � = �(�;K); � = �(�;K) andr0 = r0(�;K) with the following property: If C � X is a convex subset, R > 0 andqd : BR(0) ! X is a (K; �)-quasi-disk such that dC(qd(0)) � � then for every r 2[r0; R] the quasi-disk qd(Br(0)) is not contained in the �r+dC(qd(0))-neighborhood ofC. (Thus, qd(BR(0)) is linearly divergent from C.)Proof: It is enough to prove the following assertion: There exist positive numbersD;R such that for any quasi-disk qd : BR(0)! X, whose center qd(0) lies at distanceat least D away from a convex set C � X, there is a point q 2 qd(BR(0)) withdC(q) � 1 + dC(qd(0)).Assume that the assertion is not true. Then we have a sequence of convex setsCn, sequences of positive numbers (Rn) and (Dn) tending to in�nity and a sequenceof quasi-disks qdn : BRn ! X satisfying:dCn(qdn(0)) � Dn and dCn jqdn(BRn (0)) � 1 + dCn(qdn(0))We pick ��2n := min(Rn; Dn) and form the ultralimit X! of the sequence of basedmetric spaces (�n �X; qdn(0)). The sequence of quasi-disks yields a bilipschitz at B inX!. According to Lemma 2.3, the ultralimit of the functions �n � (dCn � dCn(qdn(0)))is the Busemann function B� associated to an ideal boundary point � of X!. Byconstruction, B� is nonpositive on B. This contradicts Lemma 3.13. 2As a consequence we see: If the boundary of a quasi-disk lies close (relative to itsradius) to a convex set C, then most of the interior of the quasi-disk lies uniformlyclose to C. More precisely:Corollary 4.2 There is a positive constant � = �(K; �) such that every (K; �)-quasi-disk qd : BR(0)! X satis�es:qd(BR(0)) � fdC � � �Rg =) qd(BR2 (0)) � fdC � �gProof: Choose � := min( �2r0 ; �2 ). If R < 2r0, then qd(BR(0)) is contained in the �-neighborhood of C. Assume that R � 2r0 and there is a point p 2 BR2 (0) withdC(qd(p)) > �. Since R2 � r0, the previous lemma implies that the quasi-diskqd(BR2 (p)) � qd(BR(0)) is not contained in the �R2 � �R-neighborhood of C, acontradiction. 215

4.2 Rigidity of quasi-isometriesLet qf : R2 ! X be a (K; �)-quasi-isometric embedding.De�nition 4.3 We call a quasi- at Q := f(R2) � X asymptotically at, if for somesequence of scale factors �n ! 0 and some base point q0 2 Q, Q! = !- lim(�n �Q; q0)is a at in the asymptotic cone X!.Proposition 4.4 Let �(K; �) be as in Lemma 4.1. If the (K; �)-quasi- at Q is asymp-totically at, then it is contained in the �(K; �)-neighborhood of a at F .Proof: By Lemma 3.8, each at F! in X! is represented by a sequence (Fn) of ats inX. If Q! = F!, we have qf(B��1n (0)) � N����1n (Fn) for !-all n. Corollary 4.2 impliesthat qf(B��1n =2(0)) � N�(Fn) for !-all n. Consequently the ats Fn subconverge to a at F which contains Q in its �-neighborhood . 2Corollary 4.5 The following properties are equivalent for (K; �)-quasi- ats Q in X:1. Q is asymptotically at.2. Q is contained in the �(K; �)-neighborhood of a at.3. Q is contained in a tubular neighborhood of a geometric component.Proof: We already proved the implication 1 =) 2. 2 =) 3 holds, because ats arecontained in geometric components. Assume that Q satis�es property 3. Then theasymptotic cone Q! is a bilipschitz at which is contained in a geometric componentof X!. Lemma 3.1 implies that Q! is a at. 2Note that if the quasi- at Q is contained in the �-neighborhood of the at F ,then Q and F have �nite Hausdor�-distance bounded in terms of the quasi-isometryconstants, cf. Corollary 2.6.We now can control the e�ect of quasi-isometries � : X ! X 0 on ats F � X.Although quasi- ats in X 0 are in general not Hausdor�-close to a at, we have:Theorem 4.6 Suppose that � : X ! X 0 is a quasi-isometry. Then the image under� of any at F in X lies within uniformly bounded Hausdor� distance from a at F 0in X 0.Proof: We proved in Lemma 3.11 that the induced bilipschitz homeomorphism �! :X! ! X 0! maps ats to ats. Hence �(F ) is an asymptotically- at quasi- at in X 0.Thus Corollary 4.5 implies that �(F ) is Hausdor�-close to a at. 2Let �#(F ) denote a at F 0 � X 0 which is Hausdor�-close to �(F ). Note that F 0is essentially unique, any other at with the same property is parallel to F 0.other. �(Fi).Lemma 4.7 Let F1; F2; F3 be pairwise nonparallel isolated ats in X which do notseparate each other. Then the ats F 01 = �#(F1); F 02 = �#(F2); F 03 = �#(F2) also donot separate each other. 16

Proof: For any r > 0 we can connect F2 and F3 outside the r-neighborhood Nr(F1)by a curve . If r is chosen su�ciently large, then the image �( ) lies on one side of�#(F1). Therefore �#(F2) and �#(F3) lie on the same side of �#(F1). 2Corollary 4.8 Let F be a boundary at in X. Then �#(F ) is a boundary at in X 0as well.Lemma 4.7 implies our Main Theorem 1.1.in the component �(Y ). hyperbolic).4.3 Structure of quasi- atsIn this section we will completely describe the quasi- ats in X. Asymptotically atquasi- ats were treated in Corollary 4.5. Let us start by constructing examples ofquasi- ats which are twisted, i.e. not asymptotically at: Take a chain S0; : : : ; Sk ofsuccessive Seifert components in X. They are separated by a chain of consecutiveisolated ats F1; : : : ; Fk where Fi = Si�1 \ Si. For 0 < i < k, there is a vertical atstrip Vi � Si which connects and is orthogonal to the successive ats Fi; Fi+1: Vi is aunion of Seifert �bers in Si and can be described as the union of all shortest geodesicsegments whose endpoints lie in Fi, respectively Fi+1. Finally we take two vertical at half-planes H0 � S0, Hk � Sk which are orthogonal to and whose boundary lineis contained in F1, respectively Fk. Note thatA := H0 [ F1 [ V1 [ F2 [ : : : [ Fk [Hkhas �nite Hausdor� distance < d from its convex hull, and A, equipped with the pathmetric, is (1; L)-quasi-isometrically embedded in X, where the positive constants d; Ldepend on the geometry ofM . Each at Fi contains a pair of distinguished transversallines arising as intersection with adjacent strips or half-planes. They divide Fi into4 quadrants. Remove from each at Fi one pair of opposite open quadrants. Whatremains from A is a quasi- at whose quasi-isometry constants are uniformly boundedin terms of k and the geometry of M .Let Q = qf(R2) be a twisted (K; �)-quasi- at. Based on our analysis of thestructure of bilipschitz ats in X!, cf. Section 3.5, we will show that Q is uniformlyclose to one of the model quasi- ats just constructed.De�nition 4.9 We say that a at F essentially splits the (K; �)-quasi- at Q if Qcontains points at distance � � = �(K; �) on the both sides of F . Otherwise wesay that Q lies essentially on one side of F . A set A essentially contains Q if Q iscontained in the �-neighborhood of A.Since Q is twisted, Corollary 4.5 implies that there are isolated ats which essen-tially split Q. We denote by Q! � X! the bilipschitz at represented by the constantsequence (Q). (Here we consider ultralimits with a constant sequence of base points.)Due to the Divergence Lemma 4.1, a at F essentially splits Q in X, if and only ifthe at F! := (F ) essentially splits Q! = (Q) in X!. According to our discussion inSection 3.5, there are �nitely many consecutive isolated ats which essentially splitQ!. Consequently, the collection of all isolated ats essentially splitting Q is �nite17

and forms a chain F1; : : : ; Fk of consecutive isolated ats in X. There is a chain ofconsecutive Seifert components S0; : : : ; Sk such that Fi = Si�1 \ Si. Their union Zis a convex set which essentially contains Q and therefore Q! � Z! = (Z). Q! isthe union of pairs of opposite quadrants in the ats Fi! and half-planes H0! � S0!,Hk! � Sk!. (Any two successive isolated ats Fi! have a line in common and the ver-tical strips inbetween therefore degenerate.) We represent the half-planes H0!; Hk!by sequences of half-planes H0n � S0, respectively Hkn � Sk, which are orthogonal toand whose boundary line is contained in F1, respectively Fk. We denote by Cn � Xthe convex hull of H0n [ Ski=1 Fi [ Hkn. Since Q! is contained in C! = (Cn), weconclude using Corollary 4.2 that for all q 2 Q and R > 0 there is a !-large set ofvalues n such that: Q \ BR(q) � N�(Cn)Observe that Q contains points in S0; Sk which are arbitrarily far away from theboundary ats F1; Fk. It follows that the sequences (H0n); (Hkn) subconverge to half-planes H0; Hk. We denote by Vi � Si, 0 < i < k, the vertical strips orthogonal toFi; Fi+1. The set A := H0 [ F1 [ V1 [ F2 [ : : : [ Fk [Hkis uniformly Hausdor�-close to its convex hull, and we conclude from the previousdiscussion that Q is contained in a uniformly bounded tubular neighborhood of A.After replacing Q by a quasi at at uniformly bounded Hausdor� distance, we mayassume that Q is contained in A. Moreover Q is a (K 0; �0)-quasi at in A equippedwith the path metric, with K 0; �0 depending on K; � and M , because the path metricon A and the metric induced from X are (1; L)-quasi-isometric with a constant L =L(M). The intersection lines with adjacent strips or half-planes divide each at Fiinto four quadrants. By Lemma 2.5, each of these quadrants is either contained inthe r-neighborhood of Q or the intersection of the quadrant with Q is r-close toits boundary, with a constant r = r(K 0; �0;M). It follows from the description ofbilipschitz ats in X! that for each Fi exactly two quadrants are contained in ther-neighborhood of Q. Similarly, the half-planes H0 and Hk are contained in ther-neighborhood of Q. This concludes the proof of:Theorem 4.10 (Classi�cation of quasi- ats) There is a constant d = d(K; �;M)so that each (K; �)-quasi- at lies at Hausdor� distance at most d from a at or atwisted model quasi- ats as described in the beginning of this section.Corollary 4.11 (1) Any (K; �)-quasi- at Q in X lies within uniform distance from a�nite union of ats. (2) The number of necessary ats is uniformly bounded in termsof K. (3) The limit set of Q in the ideal boundary @geoX is a simple loop which iscontinuous with respect to the Tits metric. (4) There is a constant K0 = K0(M) > 1such that if K � K0 then Q is asymptotically at.Proof: The �rst and third claim follow directly from our previous discussion.The asymptotic cone Q! is isometric to a complete Euclidean cone over a circle oflength l � 2(� + k�) where k is the number of isolated ats essentially separating Qand � > 0 is the minimal possible angle of intersection between the �bers of adjacentSeifert components. There is a K-bilipschitz homeomorphism [ : R2 ! Q! and weassume without loss of generality that [ maps the origin to the tip of the cone Q!.18

Let be the image of the unit circle. has length at most 2�K and circumvents thedisk of radius K�1 centered at the tip of Q!. Therefore 2�K � 2(�+k�)K andk � (K2 � 1)�� :This implies the second and fourth claim. 25 Groups quasi-isometric to fundamental groupsof Haken manifolds5.1 Quasi-actions of groups on metric spacesSuppose that � is a group and � is a map from � to the set of all (K; �)-quasi-isometriesof a metric space X.De�nition 5.1 We call � a quasi-action or under-representation of � on X if forsome constant L and all 1; 2 2 � the quasi-isometries �( 1 2) and �( 1) � �( 2) areL-close. The quasi-action is called quasi-transitive if for some constant M all orbits�(�)�x are M-close to X. The kernel (or under-represented subgroup) of the action �is the subgroup of � which consists of elements whose action on X is Hausdor�-closeto the identity. A quasi-action is called properly discontinuous if for each boundedsubset C � X there are only �nitely many elements j 2 � so that �( j)(C)\C 6= ;.To simplify notations, we will denote �( ) � x by x.A typical example of properly discontinuous quasi-transitive quasi-actions appearsas follows: Assume that the �nitely generated group � is quasi-isometric to a metricspace X, i.e. there is a quasi-isometry q from a Cayley graph of � to X. Then qtransfers the isometric action of � on the Cayley graph to a quasi-action on X. If�, equipped with a word metric, can be injectively and quasi-isometrically embeddedinto X, then there is an honest action of � on X by quasi-isometries with uniformconstants. This is the case if X is a geodesic metric space (and � in�nite).We need the next lemma for decomposing quasi-actions on trees of spaces. Let Abe a collection of subsets A � X such that:� Every bounded subset B � X intersects only �nitely many sets in A.� Any two distinct sets in A have in�nite Hausdor� distance.� There is a constant H such that for all 2 � and A 2 A the set A is H-Hausdor� close to another set in A.In this situation, we can speak of the stabilizer in � of a set A 2 A: it consists ofall elements 2 � such that A and A have �nite Hausdor� distance. Clearly thestabilizer is a subgroup of �.Lemma 5.2 If the quasi-action � is quasi-transitive then the stabilizer of any setA 2 A acts quasi-transitively on A, i.e. orbits of points in A are uniformly close toA. 19

Proof: Let B � X be a ball so that X = � � B. By assumption, only �nitely manysets 1A; : : : ; lA 2 A intersect B. Let C = maxfd( j � �1i (B); B) : 1 � i; j � lg.For x; y 2 A there are x; y 2 � so that x(x); y(y) 2 B. xA and yA are Hausdor�close to some subsets iA and jA respectively. Then xy := �1y j �1i x is in thestabilizer of A and carries x uniformly close to y: d( xy(x); y) � C + diam(B) 25.2 Quasi-actions on geometric componentsWe �rst consider the case of hyperbolic components. Let Y be the universal cover ofa hyperbolic component of M and suppose that we have a quasi-transitive action ofa group G on Y by (K; �)-quasi-isometries. Richard Schwartz [Sch] proves:� The group G �ts into a short exact sequence1 �! Fin(G) �! G �! �G �! 1 (2)with Fin(G) �nite and �G a nonuniform lattice in Isom(H 3). Hence Fin(G)is the unique maximal �nite normal subgroup of G and �G is the fundamentalgroup of a compact hyperbolic 3-orbifold with at boundary.� If F is a boundary at of Y then the quasi-action of the stabilizer of F in G iswithin bounded distance from an isometric action of a Euclidean lattice on F �=R2 . The stabilizers of boundary ats in G correspond to peripheral subgroupsof the orbifold fundamental group. Fin(G) is also the unique maximal �nitenormal subgroup of the stabilizer of F .Remark 5.3 It is unknown whether a group G satisfying (2) admits a torsion-freesubgroup of �nite index.Now we turn to the case of Seifert components. Let S = � � R be the universalcover of a Seifert component of M with hyperbolic base orbifold and consider aproperly discontinuous quasi-transitive quasi-action action of a group G on S by(K; �)-quasi-isometries. � is a convex domain of the hyperbolic plane whose boundaryis a non-empty union of disjoint geodesics. For our purposes, we are interested in thecase when the collection of boundary ats of S is invariant under this action, i.e.boundary ats are carried to within uniformly bounded distance of boundary ats.Using re ections in faces of S we extend this quasi-action to a properly discontinuousquasi-transitive quasi-action of a bigger groupH on H 2�R by (K 0; �0)-quasi-isometries;the new constants K 0; �0 depend on K; � and the geometry of �. The convex domainS � H 2 � R is quasi-preserved by G.Proposition 5.4 Any (K; �)-quasi-isometry � of a Seifert component S = � � Rquasi-preserves the Seifert �bration, i.e. there is a number r = r(K; �) such that forany s 2 � the image �(fsg � R) is r-Hausdor� close to another �ber f��(s)g � R.Proof: Any �ber fsg� R is the intersection of two orthogonal ats F; F 0 in S and ac-cording to Corollary 4.5 the images of F; F 0 are Hausdor�-close to ats �#(F ); �#(F 0)in Y . For any R > 0, the intersection of tubular neighborhoods NR(�#(F )) \20

NR(�#(F 0)) is a union C 0 � R of Seifert �bers. The diameter of C 0 is bounded abovein terms of K; � and R, because F and F 0 are orthogonal. The assertion follows. 2This Proposition was �rst proven by E. Rie�el in [R] who used quite di�erentarguments.As a consequence the quasi-action of H on H 2 � R descends to a quasi-transitivequasi-action on the hyperbolic plane by quasi-isometries with bounded constants.Let K be the kernel of this quasi-action and �H = H=K. The induced action of �Hon the ideal circle @geoH 2 by homeomorphisms is e�ective and a convergence groupaction in the sense of Gehring and Martin. Moreover, it satis�es the \simple axiscondition" of Tukia and is topologically conjugate to an action of a Moebius groupby Theorem 6B in [Tu]. This Moebius group acts cocompactly on H 2 and also properlydiscontinuously, because it preserves a locally �nite pattern of geodesics. This impliesthat the group �G = G=K is the fundamental group of a compact 2-dimensionalhyperbolic orbifold O with boundary. We therefore have an exact sequence1 �! K �! G �! �1(O) �! 1Peripheral subgroups of �1(O) correspond to stabilizers of boundary geodesics of �.Now we want to determine the structure of the kernel K of the quasi-action onhyperbolic plane. K stabilizes each �ber (up to uniformly bounded distance) andacts quasi-transitively and properly discontinuously on each �ber.Lemma 5.5 K has a unique maximal �nite normal subgroup Fin(G) and the quo-tient group K=Fin(G) is isomorphic to Z or the in�nite dihedral group D1.Proof: There is an element k 2 K which is far from the identity and preserves theorientation of the �bres on the large scale. k is quasi-isometrically conjugate to atranslation and generates an in�nite cyclic subgroup of K. The subgroup hki �= Zhas �nite index in K because K acts properly discontinuously on �bers. This impliesassertion of the lemma. 2Since �1(O) does not have nontrivial �nite normal subgroups, Fin(G) is alge-braically characterized as the unique maximal �nite normal subgroup of G. Thequotient group �G := G=Fin(G) �ts into an exact sequence1 �! Z or D1 �! �G �! �1(O) �! 1and is isomorphic to the fundamental group of a Seifert orbifold. The peripheralsubgroups of the Seifert orbifold correspond to the stabilizers of boundary ats ofS. Fin(G) is also the unique maximal �nite normal subgroup of the stabilizers ofboundary ats in G.5.3 The general caseSuppose that � is a �nitely generated group which is quasi-isometric to the universalcover X of a Haken manifold M with nontrivial canonical decomposition. We canassume without loss of generality that M is nonpositively curved. Let T be thesimplicial tree dual to the geometric decomposition of X. We have a quasi-transitiveproperly discontinuous quasi-action of � on X. By Theorem 1.1 this action induces an21

action of � by automorphisms on the tree T . The quotient T=� is a �nite graph and �therefore decomposes as a �nite graph of groups. The vertex and edge stabilizers weredescribed in section 5.2. The unique maximal �nite normal subgroups of all vertex andedge stabilizers coincide and therefore coincide with the kernel Fin(�) of the actionof � on T . The vertex stabilizers for the action of �� := �=F in(�) are fundamentalgroups of 3-dimensional hyperbolic and Seifert orbifolds with at boundary. We recallthat the edge stabilizers are peripheral subgroups of these orbifolds. We glue theseorbifolds along boundary components according to the graph T=�. The fundamentalgroup of the resulting orbifold O is isomorphic to ��. The orbifold O is �nitely coveredby a Haken manifold, cf. [MM]. This proves Theorem 1.2.References[Ba] W. Ballmann, Lectures on spaces of nonpositive curvature, DMV-Seminarvol. 25, Birkh�auser 1995.[DW] L. Van den Dries and A.J. Wilkie, On Gromov's theorem concerning groupsof polynomial growth and elementary logic, J. of Algebra, Vol. 89 (1984) 349{374.[Ge] S. M. Gersten, Divergence in 3-manifold groups, Geometric and FunctionalAnalysis, vol. 4, no. 6 (1994).[Gr1] M. Gromov, In�nite groups as geometric objects, Proc. ICM Warszawa, Vol.1 (1984) 385{ 392.[Gr2] M. Gromov, Asymptotic invariants of in�nite groups, in \Geometric GroupTheory", Vol. 2; London Math. Society Lecture Notes, 182 (1993), CambridgeUniv. Press.[JS] W. Jaco and P. Shalen, Seifert �bred spaces in 3-manifolds, Mem. Amer.Math. Soc. 220 (1979).[J] K. Johannson, Homotopy equivalences of 3-manifolds with boundary, SpringerLNM 761 (1979).[Ka] M. Kapovich, Hyperbolic Manifolds and Discrete Groups: Notes on Thurs-ton's Hyperbolization, University of Utah Lecture Notes, 1995.[KL1] M. Kapovich and B. Leeb, Actions of discrete groups on nonpositively curvedspaces, Math. Annalen, to appear.[KL2] M. Kapovich and B. Leeb, On asymptotic cones and quasi-isometry classes offundamental groups of nonpositively curved manifolds, Geometric and Func-tional Analysis, vol. 5, no. 3 (1995), 582{603.[KL3] M. Kapovich and B. Leeb, 3-manifold groups and nonpositive curvature,IHES Preprint no. M/96/47, July 1996.[KlL] B. Kleiner and B. Leeb, Rigidity of Quasi-isometries for Symmetric Spacesand Euclidean Buildings, SFB Preprint no. 457, Bonn, April 1996.22

[L] B. Leeb, 3-manifolds with(out) metrics of nonpositive curvatures, InventionesMathematicae, vol. 122 no. 2 (1995), 277{289.[LS] B.Leeb and P.Scott, A Geometric Characteristic Splitting in all Dimensions,Preprint, May 1996.[MM] D. McCullough and A. Miller, Manifold covers of 3-orbifolds with geometricpieces, Topology and its Applications 31 (1989), 169{185.[O] J.-P. Otal, Le th�eor�eme d'hyperbolisation pour les vari�et�es �br�ees de dimen-sion trois, Preprint of ENS de Lyon, 1994.[R] E. Rie�el, Groups coarse quasi-isometric to H 2�R, PhD Thesis, UCLA, 1993.[S] P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc. 15 (1983),401-487.[Sch] R. Schwartz, The quasi-isometry classi�cation of hyperbolic lattices, preprint,1994.[Th] W. Thurston, Hyperbolic structures on 3-manifolds, I. Annals of Math. 124(1986), 203- 246.[Tu] P.Tukia, Homeomorphic conjugates of fuchsian groups, J. Reine Angew.Math. 391 (1988), 1{54.Michael Kapovich, Department of Mathematics, University of Utah, Salt Lake City,UT 84112, USA; kapovich@math.utah.eduBernhard Leeb, Mathematisches Institut, Universit�at Bonn, Beringstr. 1, 53115 Bonn,Germany; leeb@rhein.iam.uni-bonn.de

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AbstractWe prove quasi-isometry invariance of the canonical decomposition for fundamentalgroups of Haken 3-manifolds with zero Euler characteristic. We show that groupsquasi-isometric to Haken manifold groups with nontrivial canonical decompositionare �nite extensions of Haken orbifold groups.1991 Mathematics Subject Classi�cation: 20F32, 53C21, 57M50.

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