3-manifold groups and nonpositive curvatureMichael Kapovich and Bernhard Leeb�August 19, 1997Abstract. We prove that the fundamental group of any compact Haken manifold ofzero Euler characteristic, which is neither Nil nor Sol, is nonpositvely curved on thelarge scale.1 IntroductionThurston's Hyperbolization Theorem provides metrics of constant negative curvatureon closed atoroidal Haken 3-manifolds. More generally, Thurston's GeometrizationConjecture asserts that all closed 3-manifolds admit a canonical minimal decompo-sition into geometric pieces. In this paper, we are interested in closed irreducible3-manifolds M with in�nite fundamental group. They topologically decompose intoSeifert and atoroidal components and, if the decomposition is non-trivial, these com-ponents can be equipped with geometric structures locally modelled on the nonpos-itively curved geometries H 2 � R, respectively H 3 . In a large number of cases suchmanifolds M admit metrics of nonpositive sectional curvature, for instance, if M isnot a graph manifold [L]. Many graph manifolds admit metrics of nonpositive curva-ture, but not all of them do [L, KL2, BK]. Another indication for the link betweenHaken manifolds and nonpositive curvature is the existence of automatic structureson the fundamental groups of all Haken manifold which are not Nil- or Sol-manifolds[E]. In particular, they admit a bicombing satisfying a fellow traveller property whichis a weak version of the convexity of distance function for nonpositively curved spaces.The aim of this paper is to establish a stronger connection between Haken manifoldsand nonpositive curvature; we show that fundamental groups of non-geometric Hakenmanifolds are nonpositively curved on the large scale in the following sense:Theorem 1.1 Let M be a Haken manifold of zero Euler characteristic, equippedwith a Riemannian metric, which is neither Nil nor Sol. Then there exists a com-pact nonpositely curved 3-manifold N with totally-geodesic at boundary and a bilip-schitz homeomorphism between the universal covers of M and N which preserves thecanonical decomposition. In particular, the fundamental groups �1(M) and �1(N) arequasi-isometric.The theorem has several direct implications for the large-scale geometry of 3-manifoldgroups �1(M):�This research was supported by the NSF grant DMS-9306140 (Kapovich), SFB 256 and IHES(Leeb) 1

1. The fundamental groups of geometric components are undistorted (quasi-iso-metrically embedded) in �1(M). Moreover there are coarse Lipschitz retractionsfrom �1(M) to the fundamental groups of its geometric components.2. If M is not Nil or Sol then all rank-two abelian subgroups are undistorted andquasi-minimize area.3. The isoperimetric inequality for �1(M) is exponential if M is Sol, cubic if M isNil, linear if M is closed hyperbolic and quadratic otherwise. This had beenpreviously known [E].We use Theorem 1.1 in [KL3] to prove that quasi-isometries preserve the canonicaldecomposition of Haken manifold groups. This yields new quasi-isometry invariantsfor fundamental groups of Haken manifolds. Another result in this direction is aspecial case of a theorem of Schwartz: he shows that the fundamental groups of cuspedhyperbolic 3-manifolds are quasi-isometric if and only if they are commensurable [Sch].Other than that the quasi-isometry classi�cation of Haken 3-manifold groups remainsopen. For instance, the following question has not yet been answered:Question 1.2 Are the fundamental groups of all (closed) graph manifolds quasi-isometric?Commensurable groups are quasi-isometric by trivial reasons. Admitting a non-positively curved metric is a commensurability invariant for Haken 3-manifolds [KL2].Thus our theorem provides quasi-isometries between noncommensurable groups.Haken manifolds which have hyperbolic components admit nonpositively curvedmetrics [L]. Therefore we will only investigate graph manifolds. Graph-manifoldsare compact Haken manifolds with boundary of zero Euler characteristic which areobtained by gluing Seifert manifolds along boundary surfaces. We exclude from theclass of graph-manifolds Sol- and Seifert manifolds. Flip-manifolds are special graph-manifolds which are constructed as follows: Take a �nite collection of products of S1with compact oriented hyperbolic surfaces with geodesic boundary. Glue them alongboundary tori by maps which interchange the basis and �ber directions. It is easy toconstruct metrics of nonpositive curvature on ip manifolds. In section 2 of this paperwe prove that the fundamental group of any graph-manifold is quasi-isometric to thefundamental group of a ip-manifold. We also show that instead of ip-manifoldsone can use manifolds �bred over the circle. Our construction generalizes an earlierexample: Epstein, Mess and Gersten discovered independently that H 2 � R is quasi-isometric to the universal cover P SL(2; R) of the unit tangent bundle of H 2 .In section 3 we apply Theorem 1.1 to study another quasi-isometry invariant,namely divergence. We prove that the fundamental groups of all graph-manifoldshave quadratic divergence. This extends earlier results of Gersten [G1, G2] whoshows that certain Hadamard spaces have quadratic divergence. We extend theseresults to all graph-manifolds. In particular, we show that for periodic geodesics in ip-manifolds the divergence is quadratic unless the geodesic is contained in a singleSeifert component. In the latter case, the divergence is linear.2

2 Construction of quasi-isometries between graphmanifold groupsTo motivate our discussion below, we recall the construction of bilipschitz homeo-morphisms between the universal cover of the unit tangent bundle UT (H 2) of thehyperbolic plane and H 2 � R, for details see [R]. We pick a base point p0 2 H 2 andidentify the unit tangent circle UTp0(H 2) with S1. If v 2 UTx(H 2) is a unit vector,we denote by �(v) 2 UTp0(H 2) the vector obtained by parallel transporting v alongthe geodesic segment [xp0]. The map UT (H 2) ! H 2 � S1 given by v 7! (x; �(v)) isbilipschitz because the area of geodesic triangles in H 2 is linearily bounded in terms ofthe shortest side length. This bilipschitz homeomorphism lifts to the universal covers.In section 2.2 below we will give a relative version of this construction.2.1 Finite covers of graph manifoldsWe recall that graph manifolds are obtained by gluing �nitely many Seifert manifoldswith hyperbolic base orbifolds. We exclude Sol and Seifert manifolds and require thatthe gluing maps between the Seifert components do not identify (unoriented) Seifert�bers up to homotopy. We shall refer to the tori and Klein bottles separating adjacentSeifert components as splitting surfaces. The universal cover ~M of a graph-manifoldM splits as the union of universal covers of Seifert components. We call them Seifertcomponents of the universal cover. We call surfaces separating Seifert components of~M splitting ats.In this section, we construct for any graph manifold a �nite cover whose Seifertcomponents and gluing maps are as simple as possible. The type of constructions weuse are well-known, see [He, MM].Lemma 2.1 Any graph-manifold M0 has an orientable �nite cover M2 where allSeifert components are trivial circle bundles over (orientable) surfaces of genus � 2.Furthermore, we can arrange that the intersection numbers of the �bres of adjacentSeifert components are �1.Proof: Step 1. By passing to the orientable cover we may assume thatM0 is orientable.Step 2. Next we make the splitting surfaces orientable. Let K1; : : : ; Km � M0be the splitting Klein bottles. Each Ki has a neighborhood N(Ki) homeomorphicto a twisted interval bundle over Ki. The boundary @N(Ki) is a 2-torus. Let � :�1(M0) ! (Z=2Z)m be the homomorphism given by the Z=2Z-intersection numberwith the Ki and let M1 be the covering of M0 corresponding to the kernel of �. TheKlein bottles Ki lift to 2-sided tori in M1 because M0 is orientable.Step 3. Now we remove singular �bers. The components Zj of the canonicaldecomposition of M1 are Seifert manifolds with base-orbifolds Oj. The orbifolds Ojhave incompressible boundary, they are in fact hyperbolic. Fix some integer p � 7and denote by Oj the orbifold obtained by attaching a disc with one singular conepoint of order p to each boundary component of Oj. The orbifolds Oj are hyperbolicand therefore have �nite nonsingular orientable covers Qj of genus � 2. We removefrom Qj the inverse images of the inserted singular discs. The resulting surface Qj3

covers Oj so that the restriction of the covering to each boundary component hasdegree p. We have exact sequences1! Z! �1(Zj) j! �1(Oj)! 1and 1! Z! �1j �1(Qj) j! �1(Qj)! 1:The subgroup �1j �1(Qj) of �1(Zj) has �nite index and the corresponding Seifertmanifold covering Zj is homeomorphic to S1 � Qj. Therefore we can �nd a p-foldcovering S1�Qj ! S1�Qj and thus obtain a �nite covering �Zj over Zj which satis�esthe properties:1. �Zj is the product of the circle with the orientable hyperbolic surface Qj of genus� 2.2. On each boundary torus of �Zj the covering �Zj ! Zj restricts to the character-istic p� p-fold cover of a boundary torus of Zj.This implies that we can glue copies of �Zj to obtain a �nite coveringM1 of the originalgraph manifold, see [He, MM].Step 4. Consider a Seifert component Z = S � S1 of M1. For each boundarytorus Tk of Z, we denote by nk the intersection number in Tk of the �ber f of Z andthe �ber fk of the other Seifert component adjacent to Tk. We attach a disk withcone point of order jnkj to each boundary circle S \ Tk of S. The resulting orbifoldis hyperbolic, because the genus of S is at least 2, and admits a �nite coveringby a surface. We extend the corresponding �nite covering S 0 ! S to a coveringZ 0 := S 0 � S1 ! S � S1 = Z by taking the product with id : S1 ! S1. Therestriction of this covering to each boundary torus T 0k ! Tk is determined by thesubgroup of �1(Tk) generated by f and fk. We repeat this for all Seifert componentsof M1 and then we glue copies of the covers Z 0 to obtain a �nite cover M2 of M1. Byconstruction, the intersection numbers of �bres of adjacent Seifert components of M2are �1. 22.2 Quasiisometric change of gluing mapsIn this section we prove our main Theorem 1.1. Let M be a graph manifold. By thediscussion in section 2.1, we can assume thatM is oriented and all Seifert componentsZj are trivial circle bundles over orientable hyperbolic surfaces: Zj = Sj � S1. Wemay further assume that for each splitting torus the intersection numbers between�bers of adjacent Seifert components are �1.Any choice of Riemannian metrics on the Seifert components, which are not re-quired to be compatible on the splitting tori, yields a path metric on M . All thesepath metrics are bilipschitz equivalent. For our purposes, the following choice isconvenient: We put negatively curved metrics on the base surfaces Sj so that allboundary components are totally geodesic of unit length. Then we equip Zj withthe product metric so that the �bers have length one. This induces canonical a�nestructures on the boundary tori of the Seifert components and we may assume thatthe gluing maps are a�ne. 4

Let Z;Z 0 be adjacent Seifert components (which may coincide) and let T � @Z,T 0 � @Z 0 be tori which are identi�ed by an a�ne gluing map A : T ! T 0. The �rsthomology groups H1(T;Z) and H1(T 0;Z) contain distinguished elements fT and fT 0corresponding to the Seifert �bres of Z and Z 0. We consider a change of the gluingmap A : T ! T 0 of the following type: De�ne BT � SL(H1(T;Z)) to be the stabiliserof fT , and analogously de�ne BT 0. Given linear transformations sT 2 BT , sT 0 2 BT 0we replace A by the new gluing map sT 0 �A � sT . We do not modify the other gluingmaps.Proposition 2.2 Let N be the manifold obtained by performing the modi�ed gluingmap sT 0 �A�sT . Then there exists a bilipschitz homeomorphism between the universalcovers ~M and ~N which preserves their canonical decompositions. In particular, �1(M)and �1(N) are quasi-isometric .Proof: We lift the canonical splitting ofM to the universal cover and denote by � thetree dual to this decomposition of ~M . Every vertex v of � corresponds to a Seifertcomponent ~Zv which universally covers a Seifert component of M , and every edge eadjacent to v corresponds to a boundary at ~Tve � @ ~Zv covering a splitting torus ofM . For each oriented edge e = [vv0] there is an a�ne gluing map ~Avv0 : ~Tve ! ~Tv0e.The group of decktransformations stabilizing an edge e acts as an integer lattice onthe splitting at ~Tve �= ~Tv0e.The Seifert foliation of each boundary torus of a Seifert component ofM induces avertical foliation of the correponding splitting at in ~M by straight lines. The matricessT ; sT 0 yield a collection of linear transformations ~sve of the splitting ats ~Tve whichpreserve vertical lines and are well-de�ned up to integer translations. Pick a Seifertcomponent ~Zv �= ~Sv�R of ~M and let ~ve � ~Sv be the boundary geodesic correspondingto ~Tve. The product decomposition ~Tve �= ~ve � R yields natural coordinates and wecan write the a�ne map ~sve as~sve(x; t) = (x; �ve(x) + t)where �ve : ~ve ! R is an a�ne function. The collection of di�erential 1-forms d�vede�nes closed 1-forms on the boundaries of ~Sv which are periodic with respect to deck-transformations. Lemma 2.4 below allows to construct bilipschitz maps Hv : ~Zv ! ~Zvwhose restrictions to boundary ats ~Tve equal ~sve up to translations: Accordingto Lemma 2.4, there exists a smooth Lipschitz function hv : ~Sv ! R such thatdhvj@ ~Sv = �v. Hence the restriction of hv to each boundary component ~ve has thesame slope as the corresponding function �ve. We de�ne a bilipschitz homeomorphismHv of ~Zv by the formula Hv(x; t) = (x; hv(x) + t):The homeomorphisms Hv have uniform bilipschitz constant C. The restriction of Hvto each boundary at ~Tve di�ers from ~sve by a vertical translation. After enlarging Cwe may assume that this vertical translation is integral. The mappings Hv can thenbe combined to a bilipschitz map between ~M and ~N . 2The following theorem implies our main theorem 1.1 stated in the introduction.Theorem 2.3 Let M be a graph manifold equipped with a Riemannian metric. Thenthere exists a nonpositively curved ip-manifold N and a bilipschitz homeomorphism5

between the universal covers ~M and ~N which preserves their canonical decomposi-tions. As a consequence, the fundamental group of any graph-manifold M is quasi-isometric to the fundamental group of a ip-manifold.Furthermore, the fundamental group of any graph-manifoldM is quasi-isometric tothe fundamental group of a manifold �bered over the circle.Proof: Pick two boundary tori T and T 0 of Seifert components of M which are gluedvia the identi�cation map A : T ! T 0. Fixed product decompositions of the Seifertcomponents give us bases ff; bg of H1(T;Z) and ff 0; b0g of H1(T 0;Z). The elementsf and f 0 correspond to the Seifert �bres. Due to Lemma 2.1 we can assume that theintersection number between Af and f 0 is �1. Hence there exist elements sT 2 BTand sT 0 2 BT 0 so thats�1T (A�1f 0) = �b and sT 0(Af) = �b0:Then the modi�ed gluing map �A = sT 0 � A � sT satis�es the ip condition�Af = �b0 and �A�1f 0 = �b:Let N be the ip manifold obtained from M by changing all gluings in this fashion.Proposition 2.2 implies the existence of a bilipschitz homeomorphism between theuniversal covers ~M and ~N which preserves the canonical decomposition. Since wechoose the metrics on the Seifert components as explained in the beginning of thissection, the gluing maps between the Seifert components of N are isometries and theinduced path metric is nonpositively curved. However the metric we obtain will besingular along splitting tori. We smooth out the singularity using the same procedureas in [L] to get a Riemannian metric of nonpositive curvature on N . This proves the�rst assertion of the theorem.To prove the second assertion, observe that the \a�ne lines" BT � b = A�1(�f)0+Z�f and A�1(BT 0 �b0) = �f+Z�A�1f 0 inH1(T;Z) intersect at the point�f+A�1(�f 0).Therefore there are transformations sT 2 BT and sT 0 2 BT 0 so that sT (b) = A�1s�1T 0 b0.The modi�ed gluing map �A = sT 0 � A � sT then satis�es �Ab = b0. The new gluingsare compatible with the foliations of the Seifert components by surfaces (given by theproduct structure). Therefore the manifold obtained from the modi�ed gluings �bersover the circle. 22.3 Lipschitz functions on universal covers of negatively cur-ved manifoldsLet S be a smooth compact manifold with strictly negative sectional curvature andtotally-geodesic boundary (in our application it will be a hyperbolic surface). Denoteby ~S the universal cover of S. Let � be a closed smooth 1-form on @S. We denotethe pull-back of � to @ ~S by � as well.Lemma 2.4 There exists a smooth Lipschitz function h on ~S satisfying dhj@ ~S = �.Proof: We �rst extend � to a smooth 1-form � on ~S which is �1(S)-invariant. Theforms � and d� are bounded. For x; y 2 ~S, we denote by yx the geodesic from y to6

x, and by x the geodesic connecting a �xed base point p 2 ~S to x. Setg(x) := Z x �:Consider a 1-connected smooth ruled surface �xy bounded by the geodesic triangle�(p; x; y) in ~S. For instance, connect the vertex p by geodesic segments to the pointson the segment yx. The sectional curvature of the induced Riemannian metric on�xy is bounded from above by the sectional curvature of S. Since S has a negativeupper curvature bound, we have the following estimates for the area of the ruledsurface: area(�xy) � constant1and area(�xy) � constant2 � d(x; y) (1)Since g(x)� g(y)� Z yx � = Z@�xy � = Z�xy d�;we have jg(x)� g(y)� Z yx �j � area(�xy) � kd�k � constant3: (2)Furthermore, j Z yx �j � constant4 � d(x; y)and j Z�xy d�j � constant5 � d(x; y)because of (1). Therefore jg(x)� g(y)j � constant6 � d(x; y)so g is Lipschitz. The closed form dg � � on @ ~S is exact. Hence we can writedg � � = dfThe smooth function f : @ ~S ! R is Lipschitz because g is Lipschitz and � is a bounded1-form. We choose f so that it has a zero on each component of @ ~S; estimate (2)then implies that f is bounded. Fix � > 0 su�ciently small so that the nearest-point-projection �@ ~S to @ ~S is well-de�ned and distance-nonincreasing on the tubularneighborhood N�(@ ~S). Take a bump function � : [0;1) ! R so that �(0) = 1 and�(t) = 0 for t � �. We extend f to ~S byf(x) := �(d(x; @ ~S)) � f(�@ ~S(x)):Since the function f is Lipschitz and bounded, the extension f is also Lipschitz andbounded. The function h := g � f on ~S has the desired properties. 27

3 Divergence of geodesicsIn this section we apply our results to discuss another quasi-isometry invariant ofgeodesic metric spaces, namely divergence. This notion was introduced in [G1] andwe recall the de�nition for the convenience of the reader. Let X be a completegeodesic metric space with one end. Pick a point x 2 X and a positive real number r.Consider the path metric on the complement X�Br(x) of a metric ball centered at x.We de�ne f(r) as the diameter of the intersection of the metric sphere Sr(x) with theunbounded component of X �Br(x). The property that f(r) grows exponentially orpolynomially of degree d does not depend on x and is a quasi-isometry invariant forX. Analoguously, one can de�ne the divergence of a complete minimizing geodesic` : R ! X. Consider the complement of the R-ball B(R) centered at `(0) equippedwith the path metric. For each R > 0, we measure the distance div(R) between thepoints `(�R) in X n BR(`(0)) equipped with the path metric. The growth rate ofthe function div is called the divergence of `. The divergence of geodesics provides anestimate from below for the divergence of the entire space.The next two results give lower estimates for the divergence of geodesics inCAT(0)-spaces. We start with an estimate for ip-manifolds.Proposition 3.1 Let M be a ip manifold with a natural metric of nonpositive cur-vature and let ` : R ! ~M be a geodesic. Then the divergence of ` is linear if ` iscontained in the union of �nitely many Seifert components and superlinear otherwise.Proof: The divergence of geodesics contained in one Seifert component is clearlylinear. Therefore the same is true for geodesics contained in a �nite union of Seifertcomponents of ~M .Suppose that ` successively intersects in�nitely many splitting ats Fn, n 2 Z,which divide ` into subsegments. Consider a shortest curve �R which connects thepoints `(�R) outside the ball BR(`(0)). We choose successive points xn on �R so thatxn 2 Fn. Clearly, the length of the portion of �R between the ats Fn�1 and Fn+1is at least d(xn�1; xn) + d(xn; xn+1). For su�ciently large R, this portion of �R liesat distance at least R=2 from the shortest geodesic segment � connecting Fn�1 andFn+1.Lemma 3.2 d(xn�1; xn) + d(xn; xn+1) � 12p2 �R� constant.Proof: We call Y� the Seifert component between Fn and Fn�1. Let p = � \ Fn anddenote by l� the Seifert �ber of Y� containing p. Since d(xn; �) � R=2 we may assumewithout loss of generality that d(xn; l+) � 12p2R. Since the nearest point projectionof Fn to Fn+1 is a strip of uniformly bounded width � w we obtain the inequalityd(xn; xn+1) � d(xn; Fn+1) � d(xn; l+)� w � d(Fn; Fn+1) � 12p2 �R � constant. 2The lemma implies that the length of the portion of �R between Fn�1 and Fn+1is at least R=4 for su�ciently big R. This �nishes the proof. 2We now give a better lower estimate for the divergence of periodic rank-onegeodesics in arbitrary locally-compact CAT(0)-spaces. Recall that a periodic geodesichas rank one if and only if it does not bound a at half-plane.8

Proposition 3.3 Let X be a locally-compact CAT(0)-space and ` be a completegeodesic which is invariant under a cyclic group of hyperbolic isometries. If ` hassubquadratic divergence, then it bounds a at half-plane and hence has linear diver-gence.Proof: We use the same notation �R as in the proof of the previous proposition.Subquadratic divergence means that the length of �R equals �R �R2 with limR!1 �R =0. Fix h > 0. Denote by � : X ! ` the nearest-point-projection. For su�ciently largeR, we can �nd a subsegment [a1a2] � `(�R=2; R=2) of length h so that the portionof �R which projects on [a1a2] via � has length at most �RhR. Pick points bi 2 �Rwith �(bi) = ai. Let �i : [0; Li]! X be the unit speed geodesic joining ai = �i(0) tobi. We have Li � R=2. The function (t) := d(�1(t); �2(t)) is convex, monotonicallyincreasing on [0; R=2] and satis�es (0) = h; (R=2) � �RRh:Therefore h � (h) � (1 + 2�Rh) � h:The quadrilateral with vertices ai and �i(h) has three sides of length h, one side oflength � (1+2�Rh)�h and angles � �=2 at ai. We have a family of such quadrilateralsQR parametrized by R. Using the translations along `, we transport the quadrilateralsQR to a �xed compact subset of X. The Hausdor� limit as R tends to in�nity ofa convergent subsequence of the translated quadrilaterals is isometric to a square ofside-length h in R2 . Hence for each h, we obtain a at square of side-length h in Xadjacent to `. The local compactness of X implies the existence of a at half-planebounded by `. 2We resume the discussion of graph manifoldsM of nonpositive curvature. Noticethat a geodesic in the universal cover ~M bounds a at half-plane if and only if itis contained in a single Seifert component. Thus any periodic geodesic in ~M whichintersects a splitting at has rank one and hence at least quadratic divergence. Anupper bound for the divergence is easier to obtain: The divergence of any geodesicin ~M and the divergence of ~M itself is at most quadratic, see also [G2] where it isproved that the divergence is at most quadratic for fundamental groups of all graph-manifolds �bred over the circle. Therefore according to Theorem 2.3 the divergencein fundamental groups of all graph-manifolds is at most quadratic.Corollary 3.4 The fundamental group of any graph-manifold has quadratic diver-gence.This extends the following earlier results in [G1], [G2]:� The divergence of fundamental groups of graph-manifolds �bered over the circleis at most quadratic.� Let � be the once punctured torus andM be the mapping torus of a Dehn twiston �. Then the divergence of �1(M) is precisely quadratic.9

References[BK] S. Buyalo and V. Kobelskii, Geometrization of graphmanifolds: isometricstates, St. Petersburg Math. Journal, Vol. 7 (1996), N 3.[E] D. Epstein, J. Cannon, D. Holt, S. Levy, M. Paterson, W. Thurston, Word-Processing in Groups, Bartlett and Jones, Boston, 1992.[G1] S. Gersten, Quadratic divergence of geodesics in CAT(0)-spaces, GeometricAnalysis and Functional Analysis, Vol. 4, no. 1 (1994), 37{ 51.[G2] S. Gersten, Divergence in 3-manifold groups, Geometric Analysis and Func-tional Analysis, Vol. 6, no. 4 (1994), 633{ 647.[He] J. Hempel, Residual �niteness for 3-manifolds, Ann. Math. Stud., vol. 111(1987), 373{396.[KL1] M. Kapovich and B. Leeb, On asymptotic cones and quasi-isometry classesof fundamental groups of 3-manifolds, Geometric Analysis and FunctionalAnalysis, Vol. 5, no. 3 (1995), 582{603.[KL2] M. Kapovich and B. Leeb, Actions of discrete groups on Hadamard spaces,Math. Ann., Bd. 306 (1996) 341{352.[KL3] M. Kapovich and B. Leeb, Quasi-isometries preserve the geometric decom-position of Haken manifolds, Vol. 128, F. 2 (1997), p. 393{416.[L] B. Leeb, 3-manifolds with(out) metrics of nonpositive curvature, Invent.Math. Vol. 122 (1995), 277{289.[MM] D. McCullough and A. Miller, Manifold covers of 3-orbifolds with geometricpieces, Topology and its Applications, Vol. 31 (1989), 169{185.[R] E. Rie�el, Groups coarse quasi-isometric to H 2 �R, PhD thesis, UCLA 1993.[Sch] R. Schwartz, The quasi-isometry classi�cation of rank one lattices, Publ. ofIHES, vol. 82 (1996), 133{ 168.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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