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: quasi-isometric 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 2-sphere; 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 quasi-isometric and yet which cannot act on the same proper geodesic metric space, properly discontinuously and cocompactly by isometries. 1

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 n-fold product of nonabelian free groups cannot act properly discontinuously on R 2n−1. 1

This paper addresses questions of quasi-isometric rigidity and classification for fundamental groups of finite graphs of groups, under the assumption that the Bass-Serre 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 quasi-isometric to the fundamental group of G is also the fundamental group of a finite graph of coarse Poincare duality groups, and any quasi-isometry between two such groups must coarsely preserves the vertex and edge spaces of their Bass-Serre 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 n-dimensional 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

Abstract. Motivated by questions in geometric group theory we define a quasisymmetric co-Hopfian property for metric spaces and provide an example of a metric Sierpiński carpet with this property. As an application we obtain a quasi-isometrically co-Hopfian 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 bi-Lipschitz transformations. 1.

We give a local condition that implies connectivity at infinity properties for CAT(0) polyhedral complexes of constant curvature.

�������� � It is a consequence of the theorem of Stallings on groups with many ends that splittings over finite groups are preserved by quasi-isometries. In this paper we use asymptotic topology to show that group splittings are preserved by quasi-isometries in many cases. Roughly speaking we show that splittings are preserved under quasi-isometries 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 quasi-isometry and the study of the relation of large-scale 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

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

Abstract. In this paper we discuss metric cell complexes satisfying a coarse form of 2-dimensional Poincaré duality. We prove that such spaces are either Gromov-hyperbolic or have polynomial growth. As an application we prove that 2-dimensional Poincaré duality groups over commutative rings are commensurable with surface groups. 1.

We state a number of open questions on 3-dimensional Poincaré duality groups and their subgroups, motivated by considerations from 3-manifold topology.

A lower bound to the action dimension of a group Sung Yil Yoon Abstract The action dimension of a discrete group Γ, actdim(Γ), is defined to be the smallest integer m such that Γ admits a properly discontinuous action on a contractible m–manifold. If no such m exists, we define actdim(Γ) ≡ ∞. Bestvina, Kapovich, and Kleiner used Van Kampen’s theory of embedding obstruction to provide a lower bound to the action dimension of a group. In this article, another lower bound to the action dimension of a group is obtained by extending their work, and the action dimensions of the fundamental groups of certain manifolds are found by computing this new lower bound. AMS Classification 20F65; 57M60