this paper we investigate its structure. The isometric action ae of G on M induces a topological action of G on @M when

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

. If G is a finitely presented group, then H 2 (#) (G, A) vanishes for all injective Banach spaces i# the regularized homological area function satisfies the linear isoperimetric inequality. This contrasts with the known result that G is word hyperbolic i# the homological area function satisfies the linear isoperimetric inequality. A closed 3-manifold group G is hyperbolic i# H 2 (#) (G, A) vanishes for all injective Banach spaces A. A vanishing theorem is proved for the fundamental group of a closed Riemannian manifold of negative curvature. 1. Introduction. It is shown in [Ge1] that if G is a hyperbolic group 1 then H 2 (#) (G, Z) vanishes strongly (the definition of the # # -cohomology is reviewed in 2 below as well as precisely what strong vanishing means). It was left open whether this result had a converse, although this was shown to be true in some special cases of interest. 2 To state my main result I need to recall some facts about the relation module...

Abstract. Let Γ be a cocompact Fuchsian group. We calculate the lower algebraic K-theory of the integral group ring ZΓ and find an explicit formula for Ki(ZΓ), i ≤ 1, in terms of the lower K-groups of finite cyclic groups. 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. Let X be a finite 2-complex with unfree fundamental group. We prove lower bounds for the area of a metric on X, in terms of the square of the least length of a noncontractible loop in X. We thus establish a uniform systolic inequality for all unfree 2-complexes. Our inequality improves the constant in M. Gromov’s inequality in this dimension. The argument relies on the Reeb graph and the coarea formula, combined with an induction on the number of freely indecomposable factors in Grushko’s decomposition of the fundamental group. More specifically, we construct a kind of a Reeb space “minimal model ” for X, reminiscent of the “chopping off long fingers ” construction used by Gromov in the context of surfaces. As a consequence, we prove the agreement of the Lusternik-Schnirelmann and systolic categories of a 2-complex. Contents 1. Inequality for unfree complexes 2

this paper. Many of the results described in this Section are either implicit or explicit in the literature of the subject. We begin by describing some Kleinian group basics, using Maskit [15] as our general reference, and some 3-manifold basics, using Hempel [10] and Jaco [11] as general references. We also state some basic Lemmas which allow us to easily translate from one language to the other. As is common practice, we suppress the explicit choice of a basepoint when referring to fundamental groups, assuming that some convenient choice of basepoint has been made.

If Γ1 and Γ2 are limit groups and S ⊂ Γ1 × Γ2 is of type FP2 then S has a subgroup of finite index that is a product of at most two limit groups. 1

Abstract. For a finitely generated free group Fn, of rank at least 2, any finite subgroup of Out(Fn) can be realized as a group of automorphisms of a graph with fundamental group Fn. This result, known as Out(Fn) realization, was proved by Zimmermann, Culler and Khramtsov. This theorem is comparable to Nielsen realization as proved by Kerckhoff: for a closed surface with negative Euler characteristic, any finite subgroup of the mapping class group can be realized as a group of isometries of a hyperbolic surface. Both of these theorems have restatements in terms of fixed points of actions on spaces naturally associated to Out(Fn) and the mapping class group respectively. For a nonnegative integer n we define a class of groups (GV P(n)) and prove a similar statement for their outer automorphism groups.

Abstract. We prove that a finitely generated group G is virtually free if and only if there exists a generating set for G and k> 0 such that all k-locally geodesic words with respect to that generating set are geodesic.