On asymptotic cones and quasi-isometry classes of fundamental groups of 3-manifolds

On asymptotic cones and quasi{isometry classesof fundamental groups of 3-manifoldsMichael Kapovich� and Bernhard LeebJanuary 19, 19951991 Mathematics Subject Classi�cation: 20F32, 53C21, 57M50.1 IntroductionLet � be a �nitely generated group. A �nite set of generators G of � determinesa Cayley graph C(�;G). It is a metric space whose quasi-isometry class does notdepend on the chosen set of generators G. We are interested in geometric prop-erties of �, i.e. quasi-isometry invariants of its Cayley graph. Well-known exam-ples of geometric properties of �nitely generated groups include: \�nitely pre-sentable", \virtually nilpotent" (Gromov), \virtually abelian" (Gromov, Brid-son and Gersten), \word hyperbolic" (Gromov), \being a �nite extension of auniform lattice in SO(n; 1)" (Mostow, Tukia, Gabai), \being a �nite extensionof a nonuniform lattice in a rank 1 symmetric space" (Schwartz), cohomolog-ical dimension is a quasi-isometry invariant for fundamental groups of �niteaspherical complexes (Gersten).Quasi-isometries ignore the local geometry. Looking for quasi-isometry in-variants we have to understand the large-scale geometry of metric spaces. Oneaspect of it, namely the asymptotic geometry of �nite subsets of distant pointsin a metric space X is encoded in the geometry of the asymptotic cone of X .This concept has been introduced by Van den Dries and Wilkie [DW] and Gro-mov [Gr2]. Bi-Lipschitz topological invariants of the asymptotic cone of X arequasi-isometry invariants of X . Papasoglu [Pa] proves that the asymptotic coneof a group satisfying a quadratic isoperimetric inequality is simply connected.The asymptotic cone will be used in [KlL] to prove quasiisometric rigidity ofnoncompact irreducible symmetric spaces of higher rank.We study the large-scale geometry of nonpositively curved spaces X . Oneobserves that ats in X are reproduced inside the asymptotic cone, whereas neg-atively curved subspaces break up into trees. One may think of the asymptotic�This research was partially supported by the grant SFB 256 \Nichtlineare partielle Dif-ferentialgleichungen" and the NSF grant DMS-9306140 (Kapovich).1

cone of X as a higher-dimensional analogue of a metric tree. For instance, theasymptotic cone of a higher-rank symmetric space is a generalized a�ne build-ing [KlL]. We investigate the pattern of ats in the asymptotic cone of certainnonpositively curved spaces of geometric rank one (in the sense of Ballmann,Brin and Eberlein) and obtain non-trivial quasi-isometry invariants.Metrics of nonpositive curvature appear in abundance in 3-dimensional topo-logy. Thurston proved that atoroidal Haken manifolds are hyperbolic. It isshown in [L] that Haken manifolds with incompressible tori generically admitmetrics of nonpositive curvature. In the subsequent paper [KaL1] we showthat the fundamental group of every Haken manifold (which is not a Nil- orSol-manifold) is quasi-isometric to the fundamental group of a 3-manifold ofnonpositive curvature.Due to the geometrization of 3-manifolds we can apply our results aboutasymptotic cones of nonpositively curved spaces to distinguish quasi-isometrytypes of fundamental groups of 3-dimensional Haken manifolds. In Theorem 5.1we prove that if a Haken manifoldM1 contains only hyperbolic components andM2 is a nonpositively curved manifold which contains a Seifert component withhyperbolic base then �1(M1) is not quasi-isometric to �1(M2). Combining thiswith results of N. Brady, Gersten and Schwartz, one obtains a rough quasi-isometry classi�cation of fundamental groups of Haken manifolds. It follows inparticular that the existence of a Seifert (as well as a hyperbolic) component ina Haken manifold is a quasi-isometry invariant of its fundamental group.The paper is organized as follows. In Section 2 we discuss basic propertiesof nonpositively curved spaces. We describe a discrete analogon of ruled sur-faces in CAT(0)-spaces. In Section 3 we review the concept of ultralimits andasymptotic cones of metric spaces. We use ultralimits to give yet another inter-pretation of the compacti�cation of representation varieties by actions of groupson trees [Mo], [Be], [Pau]. In Section 4 we study large-scale geometric propertiesof certain nonpositively curved spaces. We show that fat geodesic triangles ina CAT(0)-space X avoid regions of strictly negative curvature. Assuming thatX is negatively curved outside a disjoint union of ats, we deduce geometricand topological properties of the asymptotic cone of X . In particular, distinctembedded 2-discs have at most one point in common. This rules out the possi-bility that X contains a quasi-isometrically embedded product of the real lineand a non-abelian free group. Examples of such CAT(0)-spaces X are given byuniversal covers of Haken manifolds obtained by gluing hyperbolic components.Another class of examples are universal covers of nonpositively curved man-ifolds arising from Thurston-Schroeder's cusp-closing construction [Schr]. InSection 5 we apply the results of Section 4 to distinguish quasi-isometry classesof fundamental groups of Haken 3-manifolds.Acknowledgements. We thank the Mathematical Institute at the Univer-sity of Bonn for its hospitality during the period when this paper was written.We are grateful to Richard Schwartz and Martin Bridson for remarks concerningthe original manuscript of this paper. 2

2 Preliminaries2.1 Elementary properties of CAT(0)-spacesLet X be a complete metric space with metric d = dX . A geodesic in X is anisometric embedding f : I ! X of an interval. A complete geodesic in X is anisometric embedding f : R ! X . We denote by [xy] a geodesic segment joiningpoints x; y 2 X , and by ]xy[ the open segment. An n-dimensional at is anisometric embedding of Rn , n � 2. X is called a geodesic space if any two pointscan be connected by a geodesic. �(x; y; z) will denote a geodesic triangle in Xwith vertices x; y; z. It is the union of geodesic segments [xy], [yz] and [zx].We de�ne the inradius IRX(�) of a triangle � in X to be the in�mum of allnumbers � so that there exists a point in X with distance at most � from allsides of �.There is a synthetic way of de�ning upper curvature bounds for geodesicspaces X via distance comparison. We are only concerned with nonpositivebounds � � 0. X is said to satisfy the CAT(�)-property, if geodesic triangles inX are not thicker than triangles in the complete simply-connected Riemannian2-manifold M2� of sectional curvature �. More precisely, let �(x; y; z) be atriangle in X and choose a triangle �(x0; y0; z0) with the same side lengths inM2� . If p; q are points on �(x; y; z) and p0; q0 are points on �(x0; y0; z0), whichdivide corresponding sides in the same ratio, thend(p; q) � d(p0; q0):In fact, it su�ces to check this property only in the case when q is a vertex. Wesay that X has local upper curvature bound � at a subset A if there is a convexsubset containing A which satis�es the CAT(�)-property.X is a metric tree if it satis�es the CAT(�)-property for arbitrary negativevalues of �. In this case all geodesic triangles degenerate to tripods. One canalso characterize metric trees as geodesic spaces where any two points can beconnected by a unique simple arc (see Lemma 4.7).We collect a few facts about CAT(0)-spaces, see e.g. [GBS] and [Ba] for de-tails. The CAT(0)-property implies that the distance function is convex. Hence,any two points can be connected by a unique geodesic. In particular, CAT(0)-spaces are contractible. If Y is a convex subset in a CAT(0)-space X , then thenearest-point-projection �Y : X ! Y is well-de�ned and distance-nonincreasing.Two complete geodesic rays r1; r2 : [0;1)! X are called asymptotic, if the dis-tance function t! d(r1(t); r2(t)) remains bounded. The set @1X of equivalenceclasses of asymptotic rays is called the ideal boundary of X .Let x be a point in the CAT(0)-space X and r1; r2 : [0; �)! X be geodesicrays emanating from x. The angle \x(r1; r2) = � between r1 and r2 is de�nedby the formula: 2 sin(�2 ) = limt!0+ d(r1(t); r2(t))t3

This limit exists, because the function t 7! d(r1(t); r2(t)) is convex. The de�ni-tion coincides with the usual one in the case of Riemannian manifolds.Lemma 2.1 Let r1; r2; r3 be rays emanating from x. Then the angles betweenthem satisfy the inequality:\x(r1; r2) +\x(r2; r3) � \x(r1; r3)Lemma 2.2 If the union of the geodesic rays r1 and r2 emanating from x is ageodesic with x as interior point, then the angle between r1 and r2 equals �.Distance comparison in the presence of an upper curvature bound yieldsangle comparison:Lemma 2.3 The angles of a geodesic triangle in a CAT(�)-space are not greaterthan the corresponding angles of a comparison triangle in the model space M2�.For a geodesic triangle � in X with angles �; �; , we de�ne the angle de�citby: de�cit(�) := � � �� � � Let x; y; z be three points in the CAT(0)-spaceX . There is a unique geodesictriangle �(x; y; z). De�ne points x0; y0; z0 by [xx0] := [xy] \ [xz], [yy0] := [yz] \[yx] and [zz0] := [zx]\ [zy]. The triangle �(x0; y0; z0) is called the open trianglespanned by x; y; z. �(x; y; z) itself is called open, if it coincides with �(x0; y0; z0).We shall need the following property of CAT(0)-spaces.Lemma 2.4 Let = [xy] [ [yz] [ [zw] be a broken geodesic in a CAT(0)-spaceX such that [xy] [ [yz] and [yz] [ [zw] are geodesics. Then is a geodesic aswell.Proof: Suppose that there are points a 2 [xy] and b 2 [zw] such that d(a; b) <d(a; y) + d(y; b). Consider the comparison triangle �(a0; y0; b0) in the Euclideanplane and the point z0 2 [y0b0] with d(y0; z0) = d(y; z). Then d(a0; z0) < d(a0; y0)+d(y0; z0), on the other hand the comparison property implies that d(a0; z0) �d(a; z) = d(a; y) + d(y; z). This contradiction proves the assertion. �2.2 Nonpositively curved metrics on 3-manifoldsLet M be a compact smooth 3-manifold. A closed smooth surface S � M iscalled incompressible if it is 2-sided, has in�nite fundamental group and theinclusion S �M induces a monomorphism of fundamental groups. A manifoldM is said to be irreducible if any smooth 2-sphere in the universal cover of Mbounds a ball. If M is irreducible and contains a closed incompressible surfacethen it is called Haken. Note that if the boundary of a Haken manifold has zeroEuler characteristic then it is incompressible.4

Remark 2.5 Our de�nition of Haken manifolds is slightly more restrictive thanthe classical one (see [JS], [J]). However it will su�ce for the purposes of thispaper.Let M be a Haken 3-manifold with boundary of zero Euler characteristic.According to [J],[JS] and [Th] there is a unique �nite union T of disjoint incom-pressible 2-tori and Klein bottles which split M into a collection of hyperbolicand maximal Seifert components. We recall the following results concerning theexistence of nonpositively curved metrics on M .Theorem 2.6 ([L],[LS]) IfM admits a Riemannian metric of nonpositive sec-tional curvature with totally{geodesic boundary, then T can be isotoped so thatT [ @M is totally{geodesic.Remark 2.7 Theorem 2.6 implies that for each component Mj of M n T theuniversal cover of Mj is convex in the universal cover of M . Hence �1(Mj) isquasi{isometrically embedded into �1(M).Theorem 2.8 ([L]) Suppose that either @M is nonempty orM nT has a hyper-bolic component. Then M admits a smooth Riemannian metric of nonpositivesectional curvature with totally{geodesic boundary such that T is totally geodesicand the sectional curvature is strictly negative on each hyperbolic component ofM n T .2.3 Straight FillingsWe recall that a ruled surface in a smooth Riemannian manifold is a smoothfamily of geodesics. It is a classical fact that the intrinsic curvature of a ruledsurface is not greater than the curvature of the ambient manifold. The goal ofthis section is to construct a discrete analogue of �lling in geodesic triangles byruled surfaces.Let � be a non-degenerate triangle in Euclidean plane.We de�ne a triangulation of � to be a decomposition of � into a �nitecollection K of Eulidean 2-simplices with disjoint interiors so that the closureof their union equals �. Note that our de�nition di�ers from the standard one:we allow interior vertices on edges of triangles in K.For a triangulation S of �, we denote by Si the i-skeleton of S. A triangula-tion T of � is called special if it can be constructed from the trivial triangulationby the following inductive procedure. There exists a �nite sequence of triangu-lations � = T0; : : : ; Tn = T of �, such that the triangulation Tk+1 is obtainedfrom Tk by adding a segment �k satisfying the properties:� At least one endpoint of �k is contained in T 0k .� The intersection of the interior of �k with T 1k is empty.5

Take now a geodesic triangle �(x; y; z) in a CAT(0)-space X . We de�ne acanonical map f : T 1 ! Xby mapping � to �(x; y; z) and requiring that the restriction of f to everysegment �k is an a�ne map. We call such a map f a straight �lling. We saythat a �lling is �-�ne if the image under f of each triangle in T 1 has diameterat most �.For each triangle � in T 2, let �(�) be the curvature of X at f(@�). We put aRiemannian metric of constant curvature �(�) on � so that it has geodesic sidesand the restriction of f to every side is an isometry. This induces a path metricon � which we denote by df .Lemma 2.9 The map f : (T 1; df jT 1)! X does not increase distances.Proof: Suppose that p and q are two points on the boundary of the same triangle� in T 2. Then the distance comparison inequality implies:df (p; q) � d(f(p); f(q))The global statement follows immediately. �We de�ne the angle de�cit of the �lling f as the sumde�cit(f) := X�2T 2 de�cit(�)Lemma 2.10 The de�cit of the straight �lling f is not greater than the anglede�cit of the triangle �(x; y; z).Proof: The angles of the triangles � are not smaller than the angles of f(�) andthe sum of angle de�cits is sub-additive with respect to triangulations:de�cit(f) := X�2T 2 de�cit(�) � X�2T 2 de�cit(f(�)) � de�cit(�(x; y; z)) �Lemma 2.11 For every interior vertex p in T 0, the sum of the angles adjacentto p is at least 2�. For every vertex p, which is an interior point of a side of �,the sum of the angles adjacent to p is at least �.Proof: Consider an interior vertex p. There is exactly one segment �k whichcontains p as an interior point. Let �1 and �2 be the sums of angles in (�; df )adjacent to p from two di�erent sides of �k. Denote by i the sums of cor-responding angles in X . For each angle � adjacent to p in (�; df ), the corre-sponding angle in X adjacent to f(p) is not greater than �. Therefore, �i � i.By Lemma 2.1 and Lemma 2.2, we conclude that �i � �. The argument forvertices on the boundary is analogous. �We now compare local curvature bounds of the spaces X and (�; df ).6

Proposition 2.12 Suppose that the �lling f : T 1 ! X is �-�ne. Let p be apoint in T 1 so that the ball B�(f(p)) satis�es the CAT(�)-property with � � 0.Then the local curvature of (�; df ) at the point p is bounded from above by �.Proof: The arguments of the proof of Theorem 15 in [Ba] remain valid forsingular spaces with piecewise constant curvature. The link condition for (�; df )is satis�ed according to Lemma 2.11. �Corollary 2.13 The geodesic space (�; df ) satis�es the CAT(0)-property.Proof: According to Theorem 7 in [Ba] it su�ces to verify that any two points in(�; df ) are connected by a unique geodesic. Suppose that p; q are points whichare connected by two distinct geodesics 1 and 2. Without loss of generality,we may assume that the interiors of ga1 and 2 are disjoint. Then 1[ 2 boundsa n-gon P which is triangulated by triangles of nonpositive curvature. Usingthe Gau�-Bonnet formula and Lemma 2.11, we conclude that the sum of anglesin P is less than (n � 2)�. On the other hand, P has n � 2 angles greater orequal to � by Lemma 2.2. �2.4 Quasi-isometries of metric spacesLet (Xj ; dj) (j = 1; 2) be a pair of metric spaces. We recall that a map f :(X1; d1) ! (X2; d2) is a quasi-isometric embedding if there are two constantsK > 0 and C such thatK�1d1(x; y)� C � d2(f(x); f(y)) � Kd1(x; y) + Cfor each x; y 2 X1. A map f1 : (X1; d1) ! (X2; d2) is a quasi-isometry if thereare two constants C1; C2 and another map f2 : (X2; d2) ! (X1; d1) such thatboth f1; f2 are quasi-isometric embeddings andd1(f2f1(x); x) � C1; d2(f1f2(y); y) � C2for every x 2 X1; y 2 X2. Such spaces X1; X2 are called quasi-isometric. Forexample, two geodesic metric spaces which admit cocompact discrete actions byisometries of the same group are quasi-isometric.A �nitely generated group � with a �xed �nite set of generators carries acanonical metric which is called the word metric. The quasi-isometry class ofthe word metric does not depend on the generating set.2.5 Bi-Lipschitz embeddings of Euclidean planesLemma 2.14 Let T be a metric tree and f : R2 ! T � R be a bi-Lipschitzembedding. Then the image of f is a at in T � R.7

Proof: The map f is closed because it is bi-Lipschitz. Consider the projectionP of f(R2 ) in the tree T . The set P is a subtree in T . Let w 2 P be anypoint which separates P . Then the line fwg � R separates f(R2 ) and thereforef�1((T nfwg)�R) is not connected. We denote the intersection f(R2 )\fwg�Rby L. The preimage f�1(L) is closed in R2 . The compact subset f�1(L)[f1gin the one-point compacti�cation S2 = R2 [f1g is homeomorphic to the subsetL [ f1g in the one-point compacti�cation of the real line fwg � R. Hence byAlexander duality H1(L [ f1g;Z) �= ~H0(R2 � f�1(L);Z) 6= 0, where we useAlexander-Spanier cohomology. Thus L = fwg � R. It follows furthermorethat w separates P in exactly two components. Hence P is homeomorphic toan interval. Since f is closed, f(R2 ) = P � R. P is a complete geodesic in Tbecause f is closed and is a homeomorphism onto its image. �Corollary 2.15 The product of a metric tree and R is not bi-Lipschitz home-omorphic to the product of two metric trees with nontrivial branching.Proof: The product of two metric trees with at least 3 ends contains three atswhich have exactly one common point. �3 Ultralimits of metric spacesLet (Xi) be a sequence of metric spaces which is not precompact in the Gromov-Hausdor� topology. One can describe the limiting behavior of the sequence (Xi)by studying limits of precompact sequences of subspaces Yi � Xi. Ultra�ltersare an e�cient technical device for simultaneously taking limits of all such se-quences of subspaces and putting them together to form one object, namely anultralimit of (Xi). We discuss this concept following Gromov [Gr2].3.1 Ultra�ltersLet I be an in�nite set. A �lter on I is a nonempty family ! of subsets of Iwith the properties:� ; 62 !.� If A 2 ! and A � B, then B 2 !.� If A1; : : : ; An 2 !, then A1 \ : : : \ An 2 !.Subsets A � I which belong to a �lter ! are called !-large. We say that aproperty (P) holds for !-all i, if (P) is satis�ed for all i in some !-large set.An ultra�lter is a maximal �lter. The maximality condition can be rephrasedas: for every decomposition I = A1 [ : : : [ An of I into �nitely many disjointsubsets, the ultra�lter contains exactly one of these subsets.8

For example, for every i 2 I , we have the principal ultra�lter �i de�ned as�i := fA � I j i 2 Ag. An ultra�lter is principal if and only if it contains a�nite subset. The interesting ultra�lters are of course the non-principal ones.They cannot be described explicitly but exist by Zorn's lemma: every �lteris contained in an ultra�lter. Let Z be the Zariski �lter which consists ofcomplements to �nite subsets in I . An ultra�lter is a nonprincipal ultra�lter, ifand only if it contains Z . For us is not important how ultra�lters look like, butrather how they work: An ultra�lter ! on I assigns a \limit" to every functionf : I ! Y with values in a compact space Y . Namely,!-lim f = !-limi f(i) 2 Yis de�ned to be the unique point y 2 Y with the property that for every neigh-borhood U of y the preimage f�1U is \!-large". To see the existence of a limit,assume that there is no point y 2 Y with this property. Then each point z 2 Ypossesses a neighborhood Uz such that f�1Uz 62 !. By compactness, we cancover Y with �nitely many of these neighborhoods. It follows that I 62 !. Thiscontradicts the de�nition of a �lter. Uniqueness of the point y follows, becauseY is Hausdor�. Note that if y is an accumulation point of ff(i)gi2I then thereis a non-principal ultra�lter ! with !-lim f = y, namely an ultra�lter containingthe pullback of the neighborhood basis of y.3.2 Ultralimits of metric spacesLet (Xi)i2I be a family of metric spaces parametrized by an in�nite set I . Foran ultra�lter ! on I we de�ne the ultralimitX! = !-limi Xias follows. Let Seq be the space of sequences (xi)i2I with xi 2 Xi. The distancebetween two points (xi); (yi) 2 Seq is given byd!�(xi); (yi)� := !-lim�i 7! dXi(xi; yi)�where we take the ultralimit of the function i 7! dXi(xi; yi) with values in thecompact set [0;1]. The function d! is a pseudo-distance on Seq with values in[0;1]. Set (X!; d!) := (Seq; d!)= �where we identify points with zero d!-distance.Example 3.1 Let Xi = Y for all i, where Y is a compact metric space. ThenX! �= Y for all ultra�lters !.The concept of ultralimits extends the notion of Gromov-Hausdor� limits:9

Proposition 3.2 Let (Xi)i2N be a sequence of compact metric spaces converg-ing in the Gromov-Hausdor� topology to a compact metric space X. ThenX! �= X for all non-principal ultra�lters !.Proof: Realize the Gromov-Hausdor� convergence in an ambient compact metricspace Y , i.e. embed the Xi andX isometrically into Y such that theXi convergeto X with respect to the Hausdor� distance. Then there is a natural isometricembedding X! = !-limi Xi ��! !-limi Y �= YSince ! is non-principal, the !-limit is independent of any �nite collection ofXi's and we get: �(X!) �\i0 [i�i0Xi = XOn the other hand X � �(X!), because ��(xi)� = x if (xi) is a sequence withxi 2 Xi converging in Y to x 2 X . Hence �(X!) = X which proves the claim.�If the spacesXi do not have uniformly bounded diameter, then the ultralimitX! decomposes into (generically uncountably many) components consisting ofpoints of mutually �nite distance. We can pick out one of these components ifthe spaces Xi have basepoints x0i . The sequence (x0i )i de�nes a basepoint x0! inX! and we set X0! := �x! 2 X! j d!(x!; x0!) <1De�ne the based ultralimit as!-limi (Xi; x0i ) := (X0!; x0!)Example 3.3 For every locally compact space Y with a basepoint y0, we have:!-limi (Y; y0) �= (Y; y0)We observe that some geometric properties pass to ultralimits:Proposition 3.4 Let (Xi; x0i )i2I be a sequence of based geodesic spaces and let! be an ultra�lter. Then X0! is a geodesic space.If the Xi are CAT(�){spaces for some � � 0 then X0! has the same uppercurvature bound �.Proof: The ultralimit of geodesic segments in Xi is a geodesic segment in X0!.Therefore X0! is a geodesic space. It remains to prove that any pair of pointsx! = (xi) and y! = (yi) in X0! can be joined by a unique geodesic. Suppose thatd!(x! ; y!) = s+ t where s; t � 0. There are points zi on the geodesic segments10

[xiyi] such that for si := di(xi; zi) and ti := di(zi; yi) we have !-lim si = sand !-lim ti = t. Hence z! := (zi) satis�es d!(x! ; z!) = s and d!(z!; y!) = t.Suppose that u! = (ui) is another point with the same property. Consider in themodel spaceM2� comparison triangles �(x0i; u0i; y0i) with the same sidelengths as�(xi; ui; yi). Let z0i be a division point on [x0iy0i] corresponding to zi on [xiyi].Since !-lim(di(xi; ui) + di(ui; yi) � di(yi; xi)) = 0, we have !-lim di(ui; zi) �!-lim dM2�(u0i; z0i) = 0 and therefore u! = z!. Thus there is a unique pointz! 2 X! with d!(x! ; z!) = s and d!(z!; y!) = t. �Corollary 3.5 Let (Xi)i2N be a sequence of geodesic spaces with upper curva-ture bounds �i tending to �1. Then for every non-principal ultra�lter ! theultralimit X! is a metric forest, i.e. every component is a metric tree.3.3 The asymptotic cone of a metric spaceLet X be a metric space and ! be a non-principal ultra�lter on N. The asymp-totic cone Cone!(X) of X is de�ned as the based ultralimit of rescaled copiesof X : Cone!(X) := X0!; where (X0!; x0!) = !-limi (1i �X; x0)The limit is independent of the chosen basepoint x0 2 X . The discussion in theprevious section implies:Proposition 3.6 1. Cone!(X � Y ) = Cone!(X)� Cone!(Y ).2. Cone!Rn �= Rn .3. The asymptotic cone of a geodesic space is a geodesic space.4. The asymptotic cone of a CAT(0)-space is CAT(0).5. The asymptotic cone of a space with a negative upper curvature bound isa metric tree by Corollary 3.5.Remark 3.7 For any metric space X the asymptotic cone Cone!(X) is com-plete [DW].Remark 3.8 Suppose that X admits a cocompact discrete action by a group ofisometries. The problem of dependence of the topological type of Cone!X onthe ultra�lter ! is open (see [Gr2]).To get an idea of the size of the asymptotic cone, note that in the most in-teresting cases it is homogeneous. We call a metric space X quasi- homogeneousif diam(X=Isom(X)) is �nite.Proposition 3.9 Let X be a quasi- homogeneous metric space.Then Cone!(X) is a homogeneous metric space for every non-principal ultra-�lter !. 11

Proof: The group of sequences of isometries Isom(X)N acts transitively on theultralimit !-limi( 1i �X) which contains Cone!(X) as a component. �Lemma 3.10 Let X be a quasi-homogeneous CAT(-1){space with uncountablenumber of ideal boundary points. Then for every nonprincipal ultra�lter ! theasymptotic cone Cone!(X) is a tree with uncountable branching. Every openset in Cone!(X) contains an uncountable discrete subset.Proof: Let x0 2 X be a basepoint and y; z 2 @1X . Denote by the geodesicin X with the ideal endpoints z; y. Then Cone!([x0; y[) and Cone!([x0; z[) aregeodesic rays in Cone!(X) emanating from x0! . Their union is equal to thegeodesic Cone! . This produces uncountably many rays in Cone!(X) so thatany two of them have precisely the basepoint in common. The homogeneity ofCone!(X) implies the assertion. �Corollary 3.11 Let Z be a compact Seifert manifold with hyperbolic base or-bifold. Then the space Cone!(�1(Z)) is the product of R and a tree with un-countable branching at every point.Proof: Let � be the fundamental group of the base orbifold of Z. If Z hasnon-empty boundary, then �1(Z) virtually splits as the product of Z and a non-abelian free group. In the case @Z = ; it was proven independently by Epstein,Gersten and Mess, that �1(Z) is quasi-isometric to Z��, see [R]. The assertionfollows from Lemma 3.10. �Applications of the asymptotic cone as a quasi-isometry invaraint are basedon the followingProposition 3.12 Suppose that f : X ! Y is a quasi-isometric embedding.Then for each non-principal ultra�lter !, f induces a bi-Lipschitz embeddingCone!( f) : Cone!(X)! Cone!(Y ).If f is a quasi-isometry then Cone!( f) : Cone!(X) ! Cone!(Y ) is a bi-Lipschitz homeomorphism.We illustrate this property in the following simple case:Proposition 3.13 Let X;Y; Z be CAT (�1) spaces which have at least 3 idealboundary points. Then R �X is not quasi-isometric to Y � Z.Proof: The spaces Cone!(X);Cone!(Y ) and Cone!(Z) are metric trees withat least 3 ends. Therefore by Corollary 2.15, the spaces Cone!(Y )�Cone!(Z)and Cone!(X)� R are not bi-Lipschitz homeomorphic. �Example 3.14 H p � H q is not quasi-isometric to H p+q�1 �R, where p; q � 2.12

3.4 Limits of isometric actions on CAT(0)-spacesIn [Mo], Morgan compacti�es the space of representations of a �nitely generatedgroup � into SO(n; 1). The ideal points of the compacti�cation are isometricactions of � on metric trees. A geometric version of this construction was givenin [Be] and [Pau]. In this paragraph, we rephrase their argument in the contextof ultralimits and generalize it to the setting of nonpositive curvature.Let Xn be a sequence of CAT(0)-spaces and �n : � ! Isom(Xn) be asequence of representations. Choose a �nite generating set G of the group �.For x 2 Xn, we denote by Dn(x) the diameter of the set �n(G)(x). Set Dn :=infx2Xn Dn(x). We assume that the sequence (�n) diverges in the sense thatlimn!1Dn = 1. Choose points xn 2 Xn such that Dn(xn) � Dn + 1=n. Forany non-principal ultra�lter !, there exists a natural isometric action �! of �on the ultralimit of rescaled spaces(X!; x!) := !-limn (D�1n �Xn; xn):X! is a CAT(0)-space and the action �! has no global �xed point. If thespaces Xn are CAT(-1), then the limit space X! is a metric tree. The treeconstructed in [Be] and [Pau] is the minimal invariant subtree. Assume alsothat the spaces Xn are Hadamard manifolds of uniformly bounded dimensionwith sectional curvature bounded between two negative constants �a2;�1 andthat the representations �n are discrete and faithful. Then the Margulis lemmaimplies that the action �! is small. This means that the stabilizer of any non-degenerate segment in X! is virtually nilpotent.4 The large-scale geometry of certain CAT(0)-spaces4.1 Fat triangles in CAT(0)-spacesConsider a Haken 3-manifoldM equipped with a metric of nonpositive curvatureas in Theorem 2.8. In this section we will assume that M has at least onehyperbolic component. Let � > 0 be such that the components of T [ @M are7�-separated. Denote by �N the 3�-neighborhood of the union of T [ @M andall Seifert components of M . Then there is a negative constant � such thaton every 2�-ball with center outside �N the sectional curvature is bounded fromabove by �. The lift N of �N to the universal cover ~M consists of �-separatedconvex sets Ni. After rescaling, we can assume that � = �1.More generally, we consider a CAT(0)-space X equipped with a collectionof disjoint open convex sets Ni which satisfy the property:(�) There exists � > 0 such that each ball of radius 2� centered at a point xoutside N := [iNi is CAT(-1). 13

Denote by H the complement of N in X . Consider a geodesic triangle�(v1; v2; v3) in X and choose an �-�ne straight �lling f : T 1 ! X of thistriangle. We denote by � the CAT(0)-space (�; df ) constructed in section 2.3.Put R = R(�) := 4��1 + 2�. De�ne C� to be the set of all points in � whichhave distance not greater than R from two di�erent sides of �. The reader maythink of C� as the union of corners of the triangle �. (See Figure 1.)

C*

x

σl

γk

C*

C*Figure 1: Corners of the triangle �.Lemma 4.1 The set f((� n C�) \ T 1) is contained in N .Proof: Suppose that x is a point in (� n C�) \ T 1. Consider the concentricmetric circles k in � centered at x with radii k� for all odd numbers k so thatk� � R � �. There are L := [R=(2�)] such circles. These circles meet at mostone side of �. Suppose that each circle k contains a point xk 2 f�1(H) � T 1.The discs D�(xk) � � of radius � centered at xk are disjoint. Since the �llingf is �-�ne, every disc D�(xk) is covered by triangles �i which are contained inD2�(xk). According to Lemma 2.9, every triangle f(@�i) is contained in the ballB2�(f(xk)). By construction of � and by the property (�), the curvature �(�i)of the interior of each triangle �i is at most �1. For a measurable subset Y � �,we de�ne the integralZY (�K�)dvol := X�2T (2) ZY \�(��(�))dvol14

Using the Gau�-Bonnet formula, we estimate:de�cit(f) = Z�(�K�)dvol >Xk ZD�(xk)(�K�)dvol �L��22 > (R� 2�)��24� � �Here we use the fact that D�(xk) contains a half-disc and hence its area is atleast half the area of the Euclidean disc of radius �. On the other hand, it followsfrom Lemma 2.10 thatde�cit(f) � de�cit(�(v1; v2; v3)) � �:This contradiction implies that for at least one circle k, the intersection k\T 1is entirely contained in f�1(N). Any point on k is at distance at most �=2 froma point in k \ T 1. Therefore consecutive points of k \ T 1 are at most � apart.Since the convex subsets Ni are �-separated, f( k \ T 1) lies in one componentNi.We conclude the proof by showing that the convexity of Ni and the straight-ness of the �lling f imply: f(Dk�(x) \ T 1) � NiWe abbreviate D := Dk�(x). The intersection of k with @� is either empty orconsists of the endpoints of a subsegment � of a side of �. f(�) is containedin Ni, because Ni is convex. Recall that the triangulation T is obtained bysuccessively adding segments �l, see Section 2.3. We proceed by induction on l.Suppose that T 1l�1 \D � f�1(Ni). Then@(�l \D) � (T 1l�1 \D) [ ( k \ T 1) � f�1(Ni):The convexity of Ni implies that �l \D is contained in f�1(Ni). �We say that the triangle � in X is r-fat if its inradius is strictly greater thanr. For every vertex vi of � , we de�ne the r-corner Cr(vi) at vi to be the set ofpoints on � whose distance from both sides adjacent to vi is at most r. Notethat if � is r-fat, then the r-corners at its vertices are disjoint. We de�ne ther-fat part �r(�) of � to be � n [iCr(vi). Recall that R = R(�) = 4��1 + 2�.Proposition 4.2 Suppose that the triangle �(v1; v2; v3) is R = R(�)-fat. Thenthe fat part �R(�(v1; v2; v3)) is contained in a single component Ni.Proof: Let f : T 1 ! X be an �-straight �lling of �(v1; v2; v3). Denote by C�R(vi)the set of points on � whose distance in � to both sides [vivi�1] and [vivi+1] isat most R. Then f(C�R(vi) \ @�) � CR(vi). The C�R(vi) are convex subsets of� and since �(v1; v2; v3) is R-fat by assumption, they are disjoint and intersect15

at most two sides of @�. Thus, their complement F := � n [iC�R(vi) in � isconnected. By Lemma 4.1, F \T 1 is contained in f�1(N). The components Niare �-separated and the connected set F lies in a �=2-neighborhood of T 1. Weconclude that �R(�(v1; v2; v3)) � f(F \T 1) is contained in a single componentNi. �4.2 Asymptotic cones of certain CAT(0)-spacesWe keep the notations and assumptions of Section 4.1. In addition, we requirethat the sets Ni are 3�-neighborhoods of ats Fi in X .Pick a non-principal ultra�lter !. We de�ne F to be the family of all atsin Cone!(X) which arise as ultralimits of sequences (i�1 � Fj(i))i2N of ats inthe rescaled spaces i�1 �X .Proposition 4.3 The asymptotic cone Cone!(X) satis�es the properties:� (F1) Every open triangle is contained in a at F 2 F .� (F2) Any two ats in F have at most one point in common.Proof: Let � = �(x; y; z) be an open triangle in Cone!(X). Then � is theultralimit of a sequence of triangles i�1��i, where �i = �(xi; yi; zi) are trianglesin the original space X . For !-every i the triangle �i is R-fat, where R ischosen as in Section 4.1. Otherwise, the ultralimit � would not be open. ByProposition 4.2, the fat part �R(�i) is contained in a set Nj(i). Each point won the side ]xy[ of � corresponds to a sequence of points wi on ]xiyi[. Since �is open, we have:0 < d!(w; [xz] [ [zy]) = !-lim 1i � d(wi; [xizi] [ [ziyi])Hence for !-every i, wi does not belong to any R-corner of �i. Therefore, wibelongs to Nj(i) and its distance from the at Fj(i) is at most 3�. We concludethat w lies in the at F 2 F which arises as the ultralimit of the sequence(i�1 � Fj(i)). This concludes the proof of property (F1).To verify property (F2), let F and F 0 be ats in F which have two distinctpoints x and y in common. We will show that F 0 � F . Choose a point z0 inF 0 so that the triangle �(x; y; z0) is non-degenerate and pick points u and won ]xy[ and ]xz0[. There is a sequence of ats (Fj(i)) in X which correspondsto the at F . Select points xi; yi 2 Fj(i), z0i 2 X , ui 2]xi; yi[ and wi 2]xi; z0i[so that (xi); (yi); (z0i); (ui); (wi) represent the points x; y; z0; u; w. For !-all i,ui; wi belong to the fat part �R(�(xi; yi; z0i)). According to Proposition 4.2,the points ui; wi belong to the same component Nk(i). Since ui lies on Fj(i),Nk(i) coincides with Nj(i). Hence, w lies on F . We conclude that z0 2 F , sincew was an arbitrary point of ]xz0[. �16

4.3 Special CAT(0)-spacesIn the previous section, we established geometric properties for the asymptoticcone of a CAT(0)-space with isolated ats. The asymptotic cone is a CAT(0)-space itself and now we shall study geometric and topological properties ofCAT(0)-spaces Y satisfying the conclusion of Proposition 4.3.Consider a at F 2 F and denote by �F : Y ! F the nearest-point-projection onto F .Lemma 4.4 Let : I ! Y be a curve in the complement of F . Then �F � isconstant.Proof: Assume that �F � is non-constant. Then there exist nearby points p1and p2 on with distinct projections qi := �F (pi) in F :d(pi; F ) = d(pi; qi) > d(p1; p2) (i = 1; 2)The geodesic [p1p2] cannot meet F and therefore the piecewise geodesic path[p1q1q2p2] is not locally minimizing at q1 or q2, say at q1 (see Lemma 2.4).Since [p1q1] \ [q1q2] = fq1g, the triangle �(p1; q1; q2) spans a non-degenerateopen triangle �(r; q1; s). By property (F1), �(r; q1; s) lies in a at F 0. SinceF \F 0 contains the non-trivial segment [q1s], F and F 0 must coincide accordingto (F2). Thus [p1q1] \ F contains a non-trivial segment [q1r]. This contradictsthat q1 = �F (p1). �Lemma 4.5 Every embedded closed curve � Y is contained in a at F 2 F .Proof: Consider the geodesic segment � joining two distinct points x and y on . Since is a closed curve, the projection �� maps at least two points of toan interior point u of �. Hence there exists a point z on n � with ��(z) = u.Consider a maximal subarc � � containing z with ��(�) = fug. At least oneof the endpoints of � is di�erent from u, i.e. does not lie on �. Denote it by z1.There is a nearby point w on whose projection ��(w) =: v is di�erent from uand which satis�es d(w; z1) < d(u; z1) = d(�; z1)As in the proof of Lemma 4.4, we �nd a at F 2 F which contains a non-degenerate segment �0 � �.We proceed by proving that �F (x) 6= �F (y). The intersection F \ � isa non-degenerate segment [x0y0], so that x0 lies between x and y0. Considerx00 := �F (x) and suppose that x00 6= x0. Then the piecewise geodesic path xx0x00is not locally minimizing at x0. Since [xx0] \ [x0x00] = fx0g, �(x; x0; x00) spans anon-degenerate open triangle with vertex x0. As in the proof of Lemma 4.4 weobtain a contradiction. Therefore �F (x) = x0 and similarly �F (y) = y0. Thus�F � is non-constant. 17

Suppose now that 6� F . Choose a maximal open subarc � � in thecomplement of F . By Lemma 4.4, �F (�) is a point p 2 F . By maximalityof � and continuity we conclude that every endpoint of � must coincide withp. Therefore � has at most one endpoint and �F ( ) = fpg. This contradicts�F (x) 6= �F (y). We conclude that is contained in F . �Corollary 4.6 Every embedded disc in Y of dimension at least 2 is containedin a at F 2 F . In particular, there are no other ats in Y besides the atsF 2 F .We can use arguments similar to the proof of Lemma 4.4 to show:Lemma 4.7 Suppose that T is a metric tree. Then T is a topological tree, i.e.any two points are connected by a unique topologically embedded arc.We conclude from Lemma 4.5:Corollary 4.8 Suppose that Y is a CAT(0){space satisfying the conclusion ofProposition 4.3 and all ats in Y have dimension 2. Let T be a tree withnontrivial branching. Then there is no topological embedding � : T � R ! Y .Corollary 4.9 Let Y be a CAT(0){space satisfying the conclusion of Proposi-tion 4.3. Suppose that T is a metric tree which contains an uncountable discretesubset. Then there is no bi-Lipschitz embedding � : T � R ! Y .Proof: Suppose that there is such an embedding �. Lemma 4.5 and property(F2) imply that the image of � is contained in a at F 2 F . We obtain acontradiction, since a at does not contain uncountable discrete subsets. �5 Distinction of quasi-isometry classes of 3-ma-nifold groupsThe goal of this section is to distinguish quasi-isometry classes of fundamentalgroups of certain 3-manifolds. Recall that any Haken manifold of zero Eulercharacteristic can be obtained in a unique way by gluing hyperbolic and maximalSeifert components. In this section we consider only such Haken manifolds.Theorem 5.1 Let M1 be a non-positively curved Haken manifold which has atleast one Seifert component with hyperbolic base. Assume that M2 is a Hakenmanifold which contains only hyperbolic components. Then the fundamentalgroups �1(M1) and �1(M2) are not quasi-isometric.18

Remark 5.2 As we shall prove in [KaL1], the condition in Theorem 5.1 thatM1 admits a metric of non-positive curvature is actually obsolete. Namely, weprove that fundamental group of any Haken manifold which is neither Sol norNil, is quasi-isometric to the fundamental group of a 3-manifold of nonpositivecurvature.Proof: The manifold M1 contains a Seifert component Z. The universal cover~Z of Z is a convex subset in the universal cover ~M1 according to 2.6. Therefore,the asymptotic cone Cone!( ~Z) is isometrically embedded in Cone!( ~M1). Theasymptotic cone Cone!( ~Z) is isometric to the product of the real line and ametric tree T with nontrivial branching, see Corollary 3.11. Suppose that thereexists a quasi-isometry ~M1 ! ~M2. It induces a homeomorphism Cone!( ~M1)!Cone!( ~M2). Hence, R�T topologically embeds into Cone!( ~M2). The manifoldM2 carries a metric of nonpositive curvature (Theorem 2.8). By Theorem 4.3,Cone!( ~M2) satis�es the properties (F1) and (F2), see the discussion in thebeginning of section 4.1. This contradicts Corollary 4.8. �Theorem 5.3 Let M be a nonpositively curved Haken 3-manifold with totally-geodesic at boundary. Assume that M is not at, not Seifert and not home-omorphic to a closed hyperbolic manifold. Then the asymptotic cone of theuniversal cover of M contains two ats which have exactly one point in com-mon.Proof: Suppose that M contains a hyperbolic component N . By Theorem2.6, the universal cover ~N is convex in ~M . Hence, Cone!( ~N) is isometricallyembedded in Cone!( ~M). Pick two ats F1 and F2 in @ ~N . Then Cone!(F1) andCone!(F2) are ats in Cone!( ~N) which both contain the base point. Accordingto Proposition 4.3 they have exactly one common point.We are left with the case that M is a graph-manifold. We can �nd in theuniversal cover ~M two convex subsets A1 and A2 which are universal coversof Seifert components and whose intersection is a at F . The sets Ai split o�Riemannian factors li isometric to the real line. Since M is not Seifert, wemay assume that the one-dimensional factors are not parallel in F . Choose ats Fi in Ai di�erent from F and consider the associated ats Cone!(Fi) inCone!( ~M). The intersection of Cone!(Fi) with Cone!(F ) is a line Cone!( li).The lines Cone!( li) intersect in a single point. Since the intersection of the setsCone!(Ai) is precisely Cone!(F ), the ats Cone!(Fi) have exactly one pointin common. �Theorem 5.1 combined with results of Gromov, Gersten, N. Brady,Schwartz and ourselves leads to a rough classi�cation of quasi-isometry typesof fundamental groups of Haken manifolds. We divide Haken 3-manifolds with at incompressible boundary into the following classes.1. H : closed hyperbolic 3-manifolds.19

2. CH : open hyperbolic 3-manifolds of �nite volume.3. HH : manifolds not contained in H; CH which are obtained by gluinghyperbolic components only.4. S : Seifert manifolds with hyperbolic base-orbifolds.5. SS : graph-manifolds. They are obtained by gluing Seifert manifolds withhyperbolic base and they are not Seifert.6. HS : manifolds with at least one hyperbolic and Seifert component (withhyperbolic base).7. Closed Nil-manifolds.8. Closed Sol-manifolds.9. Flat manifolds.Theorem 5.4 If two 3-manifolds belong to di�erent classes (1{9) then theirfundamental groups are not quasi-isometric.Proof: The fundamental groups of Nil- and at manifolds have polynomialgrowth of degree 4 in the nilpotent and of degree at most 3 in the at case.Therefore they are not quasi-isometric to each other and to the fundamentalgroups of all other classes.The property to be word-hyperbolic is a quasi-isometry invariant [GdH].Therefore, the fundamental groups of closed hyperbolic manifolds are not quasi-isometric to the fundamental groups of manifolds of all other classes.LetM be a manifold of the class CH and � be a �nitely generated torsion{freegroup which is quasi-isometric to �1(M). Corollary 4 in the paper of R. Schwartz[Schw2] implies that � must be isomorphic to a lattice in SO(3; 1) which iscommensurable with �1(M). Therefore, if such a group � is the fundamentalgroup of a Haken 3-manifold, then � belongs to the class CH.Theorem 5.1 and Remark 5.2 imply that the fundamental groups of the classHH have di�erent quasi-isometry type from the classes HS, SS and S.Gersten introduced in [Ge1] a quasi-isometry invariant notion of divergenceof geodesics which measures the rate of growth of diameters of spheres. Using[Br], Gersten [Ge2] shows that fundamental groups of manifolds in the classesHS and HH have exponential divergence. In [Ge2] Gersten proves that thefundamental groups of all graph-manifolds �bered over the circle have at mostquadratic divergence. On the other hand, [KaL1] implies that the fundamentalgroup of any graph-manifold is quasi-isometric to the fundamental group of agraph-manifold �bered over the circle. This distinguishes the classes HS andSS . Note that Gersten characterizes closed graph{manifolds as those Hakenmanifolds whose fundamental groups have precisely quadratic divergence.20

To distinguish the fundamental groups of Seifert manifolds and manifolds inHH;HS;SS up to quasi-isometry we observe that their asymptotic cones havedi�erent topological properties. Namely, the asymptotic cone of the fundamen-tal group of a Seifert manifold with hyperbolic base splits as a metric productT � R where T is a tree with nontrivial branching, see Corollary 3.11. Hencethe intersection of bi-Lipschitz embedded Euclidean planes is either empty orcontains a line, according to Lemma 2.14. On the other hand, by Theorem 5.3,the asymptotic cones of manifolds in the classesHH;HS;SS contain ats whichhave precisely one point in common.To sever the class of Sol{manifolds one can use the fact that amenability is aquasi-isometry invariant. The only Haken manifolds with amenable fundamentalgroups are Sol-, Nil- and at manifolds. One may also argue as follows on thelevel of asymptotic cones. It was shown in [Gr2] that the asymptotic cone of theLie group Sol is not simply-connected. On the other hand, if M is a manifoldof nonpositive curvature, then the asymptotic cone of the universal cover ~M iscontractible (see 3.4). �Remark 5.5 A theorem of Rie�el [R] distinguishes quasi-isometry classes offundamental groups of closed Seifert manifolds with hyperbolic base from thefundamental groups of all other 3-manifolds.Remark 5.6 Fundamental groups of open and closed aspherical 3-manifoldscannot be quasi-isometric, because they have di�erent cohomological dimension[Ge3].Remark 5.7 The question how to distinguish quasi-isometry types of funda-mental groups inside the classes HS ;SS and HH remains open. Consider-able progress in this direction was achieved by Schwartz [Schw1] who provesthat fundamental groups of two open hyperbolic manifolds of �nite volume arequasi-isometric i� they are commensurable. We discuss in our consecutive paper[KaL2] the quasi-isometry invariance of the canonical decomposition for (uni-versal covers of) Haken manifolds of zero Euler characteristic.References[Ba] W. Ballmann, Singular spaces of nonpositive curvature, in [GdH], pp.189-201.[Be] M. Bestvina, Degenerations of hyperbolic space, Duke Math. Journal,Vol. 56 (1988) N 1.[Br] N. Brady, Divergence of geodesics in manifolds of negative curvature,in preparation. 21

[DW] L.Van den Dries, A.J.Wilkie, On Gromov's theorem concerning groupsof polynomial growth and elementary logic, Journ. of Algebra, Vol. 89(1984) 349{ 374.[Ge1] S. Gersten, Quadratic divergence of geodesics in CAT(0)-spaces, Geo-metric and Functional Analysis, Vol 4, N 1 (1994) 37{ 51.[Ge2] S. Gersten, Divergence in 3-manifolds groups, Preprint, University ofUtah, 1993.[Ge3] S. Gersten, Quasi-isometry invariance of cohomological dimension, C.R. Acad. Sci. Paris, t. 316 (1993), S�erie 316 I, p. 411-416.[GdH] E. Ghys and P. de la Harpe, Sur les groupes hyperboliques d'apr�esMikhael Gromov, Birkh�auser 1990.[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 \GeometricGroup Theory", Vol. 2; London Math. Society Lecture Notes, 182(1993), Cambridge Univ. Press.[GBS] M. Gromov, W. Ballmann, V. Schroeder, Manifolds of nonpositive cur-vature, Birkh�auser 1985.[JS] W. Jaco, P. Shalen, Seifert �bre spaces in 3-manifolds, Memoirs ofAMS, 1979, no. 2.[J] K. Johannson, Homotopy{equivalences of 3{manifolds with boundary,Springer Lecture Notes in Math., Vol. 761, 1979.[KaL1] M. Kapovich, B. Leeb, On quasi{isometries of graph-manifold groups,Preprint, 1994.[KaL2] M. Kapovich, B. Leeb, Quasi-isometries preserve the canonical decom-position of Haken manifolds, Preprint, 1994.[KlL] B. Kleiner, B. Leeb, Rigidity of quasi-isometries for symmetric spacesof higher rank, Preprint, 1995.[L] B. Leeb, 3-manifolds with(out) metrics of nonpositive curvatures, PhDThesis, University of Maryland, 1992.[LS] B. Leeb, P. Scott, Decomposition of nonpositively curved manifolds, inpreparation.[Mo] J. Morgan, Group actions on trees and the compacti�cation of the spaceof classes of SO(n, 1) representations, Topology, 1986, Vol. 25, no. 1,p. 1{33. 22

[Pa] P. Papasoglu, On the asymptotic cone of groups satisfying a quadraticisoperimetric inequality, preprint.[Pau] F. Paulin, Topologie de Gromov �equivariant, structures hyperboliques etarbres reels, Inv. Math. 94 (1988) 53{ 80[R] E. Rie�el, Groups coarse quasi-isometric to H 2�R, PhD Thesis, UCLA,1993.[Schr] V. Schroeder, A cusp closing theorem, Proc. AMS 106 (1989) no. 3,797-802.[Schw1] R. Schwartz, The quasi-isometry classi�cation of hyperbolic lattices,preprint, 1993.[Schw2] R. Schwartz, On the quasi-isometry structure of rank 1 Lattices,preprint, 1994.[Sc] P. Scott, The geometries of 3-manifolds, Bull. London Math. Soc..15(1983), 404-487.[Th] W. Thurston, Hyperbolic structures on 3-manifolds, I. Ann. of Math.124 (1986), 203- 246.Michael Kapovich, Department of Mathematics, University of Utah, Salt LakeCity, UT 84112, USA; [email protected] Leeb, Mathematisches Institut, Universit�at Bonn, Beringstr. 1, 53115Bonn, Germany; [email protected]

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