Results 1  10
of
16
Hyperbolic Groups With Low Dimensional Boundary
 Ann. Sci. École Norm. Sup
, 2000
"... If a torsionfree hyperbolic group G has 1dimensional boundary @1G, then @1G is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When @1G is a Sierpinski carpet we show that G is a quasiconvex subgroup of a 3dimensional hyperbolic Poincar'e duality group. We ..."
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Cited by 30 (9 self)
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If a torsionfree hyperbolic group G has 1dimensional boundary @1G, then @1G is a Menger curve or a Sierpinski carpet provided G does not split over a cyclic group. When @1G is a Sierpinski carpet we show that G is a quasiconvex subgroup of a 3dimensional hyperbolic Poincar'e duality group. We also construct a "topologically rigid" hyperbolic group G: any homeomorphism of @1G is induced by an element of G.
van Kampen’s embedding obstructions for discrete groups
 Invent. Math
, 2002
"... We give a lower bound to the dimension of a contractible manifold on which a given group can act properly discontinuously. In particular, we show that the nfold product of nonabelian free groups cannot act properly discontinuously on R 2n−1. 1 ..."
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Cited by 9 (2 self)
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We give a lower bound to the dimension of a contractible manifold on which a given group can act properly discontinuously. In particular, we show that the nfold product of nonabelian free groups cannot act properly discontinuously on R 2n−1. 1
Quasi–actions on trees I, bounded valence
 Annals of Mathematics
"... Given a bounded valence, bushy tree T, we prove that any quasiaction of a group G on T is quasiconjugate to an action of G on another bounded valence, bushy tree T ′. This theorem has many applications: quasiisometric rigidity for fundamental groups of finite, bushy graphs of coarse PD(n) groups fo ..."
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Cited by 9 (3 self)
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Given a bounded valence, bushy tree T, we prove that any quasiaction of a group G on T is quasiconjugate to an action of G on another bounded valence, bushy tree T ′. This theorem has many applications: quasiisometric rigidity for fundamental groups of finite, bushy graphs of coarse PD(n) groups for each fixed n; a generalization to actions on Cantor sets of Sullivan’s Theorem about uniformly quasiconformal actions on the 2sphere; and a characterization of locally compact topological groups which contain a virtually free group as a cocompact lattice. Finally, we give the first examples of two finitely generated groups which are quasiisometric and yet which cannot act on the same proper geodesic metric space, properly discontinuously and cocompactly by isometries. 1
Quasiactions on trees II: Finite depth BassSerre trees
, 2004
"... This paper addresses questions of quasiisometric rigidity and classification for fundamental groups of finite graphs of groups, under the assumption that the BassSerre tree of the graph of groups has finite depth. The main example of a finite depth graph of groups is one whose vertex and edge grou ..."
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Cited by 8 (2 self)
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This paper addresses questions of quasiisometric rigidity and classification for fundamental groups of finite graphs of groups, under the assumption that the BassSerre tree of the graph of groups has finite depth. The main example of a finite depth graph of groups is one whose vertex and edge groups are coarse Poincare duality groups. The main theorem says that, under certain hypotheses, if G is a finite graph of coarse Poincare duality groups then any finitely generated group quasiisometric to the fundamental group of G is also the fundamental group of a finite graph of coarse Poincare duality groups, and any quasiisometry between two such groups must coarsely preserves the vertex and edge spaces of their BassSerre trees of spaces. Besides some simple normalization hypotheses, the main hypothesis is the “crossing graph condition”, which is imposed on each vertex group Gv which is an ndimensional coarse Poincare duality group for which every incident edge group has positive codimension: the crossing graph of Gv is a graph ǫv that describes the pattern in which the codimension 1 edge groups incident to Gv are crossed by other edge groups incident to Gv, and the crossing graph condition requires that ǫv be connected or empty. 1
LocaltoAsymptotic Topology for Cocompact CAT(0) Complexes
, 2001
"... We give a local condition that implies connectivity at infinity properties for CAT(0) polyhedral complexes of constant curvature. ..."
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Cited by 7 (1 self)
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We give a local condition that implies connectivity at infinity properties for CAT(0) polyhedral complexes of constant curvature.
A SIERPIŃSKI CARPET WITH THE COHOPFIAN PROPERTY
"... Abstract. Motivated by questions in geometric group theory we define a quasisymmetric coHopfian property for metric spaces and provide an example of a metric Sierpiński carpet with this property. As an application we obtain a quasiisometrically coHopfian Gromov hyperbolic space with a Sierpiński ..."
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Cited by 3 (1 self)
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Abstract. Motivated by questions in geometric group theory we define a quasisymmetric coHopfian property for metric spaces and provide an example of a metric Sierpiński carpet with this property. As an application we obtain a quasiisometrically coHopfian Gromov hyperbolic space with a Sierpiński carpet boundary at infinity. In addition, we give a complete description of the quasisymmetry group of the constructed Sierpiński carpet. This group is uncountable and coincides with the group of biLipschitz transformations. 1.
GROUP SPLITTINGS AND ASYMPTOTIC TOPOLOGY
"... �������� � It is a consequence of the theorem of Stallings on groups with many ends that splittings over finite groups are preserved by quasiisometries. In this paper we use asymptotic topology to show that group splittings are preserved by quasiisometries in many cases. Roughly speaking we show t ..."
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Cited by 3 (1 self)
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�������� � It is a consequence of the theorem of Stallings on groups with many ends that splittings over finite groups are preserved by quasiisometries. In this paper we use asymptotic topology to show that group splittings are preserved by quasiisometries in many cases. Roughly speaking we show that splittings are preserved under quasiisometries when the vertex groups are fundamental groups of aspherical manifolds (or more generally ‘coarse P D(n)groups’) and the edge groups are ‘smaller ’ than the vertex groups. The notion of quasiisometry and the study of the relation of largescale geometry of groups and algebraic properties has become predominant in group theory after the seminal papers of Gromov [G1,G2]. A classical theorem of Stallings ([St]) implies, that if G splits over a finite group
Homological dimension and critical exponent of Kleinian groups
, 2007
"... We prove an inequality between the relative homological dimension of a Kleinian group Γ ⊂ Isom(H n) and its critical exponent. As an application of this result we show that for a geometrically finite Kleinian group Γ, if the topological dimension of the limit set of Γ equals its Hausdorff dimension, ..."
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Cited by 3 (1 self)
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We prove an inequality between the relative homological dimension of a Kleinian group Γ ⊂ Isom(H n) and its critical exponent. As an application of this result we show that for a geometrically finite Kleinian group Γ, if the topological dimension of the limit set of Γ equals its Hausdorff dimension, then the limit set is a round sphere. 1
Geometry of quasiplanes
, 2004
"... Abstract. In this paper we discuss metric cell complexes satisfying a coarse form of 2dimensional Poincaré duality. We prove that such spaces are either Gromovhyperbolic or have polynomial growth. As an application we prove that 2dimensional Poincaré duality groups over commutative rings are comm ..."
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Cited by 2 (1 self)
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Abstract. In this paper we discuss metric cell complexes satisfying a coarse form of 2dimensional Poincaré duality. We prove that such spaces are either Gromovhyperbolic or have polynomial growth. As an application we prove that 2dimensional Poincaré duality groups over commutative rings are commensurable with surface groups. 1.
Some questions on subgroups of 3dimensional Poincaré duality groups
"... We state a number of open questions on 3dimensional Poincaré duality groups and their subgroups, motivated by considerations from 3manifold topology. ..."
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We state a number of open questions on 3dimensional Poincaré duality groups and their subgroups, motivated by considerations from 3manifold topology.