. A 2-dimensional orbihedron of nonpositive curvature is a pair (X; \Gamma), where X is a 2-dimensional simplicial complex with a piecewise smooth metric such that X has nonpositive curvature in the sense of Alexandrov and Busemann and \Gamma is a group of isometries of X which acts properly discontinuously and cocompactly. By analogy with Riemannian manifolds of nonpositive curvature we introduce a natural notion of rank 1 for (X; \Gamma) which turns out to depend only on \Gamma and prove that, if X is boundaryless, then either (X; \Gamma) has rank 1, or X is the product of two trees, or X is a thick Euclidean building. In the first case the geodesic flow on X is topologically transitive and closed geodesics are dense. 1. Introduction The idea of considering curvature bounds on metric spaces belongs to Alexandrov [Ale], Busemann [Bus] and Wald [Wal]. Busemann initiated the theory of spaces of nonpositive curvature. Later, Bruhat and Tits [BrTi] showed that there is a natural ...

Abstract. We prove that any isomorphism θ: M0 ≃ M of group measure space II1

this paper we shall prove the coarse Baum-Connes conjecture for proper metric spaces with nite asymptotic dimension. Combining this result with a certain descent principle we obtain the following application to the Novikov conjecture on homotopy invariance of higher signatures. Theorem 1.1 Let be a nitely presented group whose classifying space B

The study of the local-global geometric topology of homology manifolds has a long history. Homology manifolds were introduced in the 1930s in attempts to identify local homological properties that implied the duality theorems satis ed by manifolds [25, 57]. Bing's work on decomposition space theory opened new perspectives. He constructed important examples of 3-dimensional homology manifolds with nonmanifold points, which led to the study of other structural properties of these spaces, and also established his shrinking criterion that can be used to determine when homology manifolds obtained as decomposition spaces of manifolds are manifolds [4]. In the 1970s, the fundamental work of Cannon and Edwards on the double suspension problem led Cannon to propose a conjecture on the nature of manifolds, and generated a program that culminated with the Edwards-Quinn characterization of higher-dimensional topological manifolds as ENR homology manifolds satisfying a weak general position property known as the disjoint disks property [17, 26,23]. Starting with the work of Quinn [45, 47], a new viewpoint has emerged. Recent advances [10] use techniques of controlled topology to produce a wealth of previously

Abstract. We give a precise definition of “generic-case complexity” and show that for a very large class of finitely generated groups the classical decision problems of group theory- the word, conjugacy and membership problems- all have linear-time generic-case complexity. We prove such theorems by using the theory of random walks on regular graphs. Contents 1. Motivation

Abstract. We give the first part of a proof of Thurston’s Ending Lamination conjecture. In this part we show how to construct from the end invariants of a Kleinian surface group a “Lipschitz model ” for the thick part of the corresponding hyperbolic manifold. This enables us to describe the topological structure of the thick part, and to give a-priori geometric bounds. Contents

2. Complexes and subcomplexes of curves 11 3. Projection bounds 19 4. Tight geodesics and hierarchies 25

2. Basic definitions and preliminaries 3 A. Adaptedness to the graph structure 4 B. Reversible Markov chains 4

Abstract. We generalize some results of Paulin and Rips-Sela on endomorphisms of hyperbolic groups to relatively hyperbolic groups, and in particular prove the following. • If G is a non-elementary relatively hyperbolic group with slender parabolic subgroups, and either G is not co-Hopfian or Out(G) is infinite, then G splits over a slender group. • If H is a non-parabolic subgroup of a relatively hyperbolic group, and if any isometric H-action on an R-tree is trivial, then H is Hopfian. • If G is a non-elementary relatively hyperbolic group whose peripheral subgroups are finitely generated, then G has a non-elementary relatively hyperbolic quotient that is Hopfian. • Any finitely presented group is isomorphic to a finite index subgroup of Out(H) for some group H with Kazhdan property (T). (This sharpens a result of Ollivier-Wise). 1.

This is the first of two papers devoted to connections between asymptotic functions of groups and computational complexity. One of the main results of this paper states that if for every m the first m digits of a real number α ≥ 4 are computable in time ≤ C2 2Cm for some constant C> 0 then n α is equivalent (“big O”) to the Dehn function of a finitely presented group. The smallest isodiametric function of this group is n 3/4α. On the other hand if n α is equivalent to the Dehn function of a finitely presented group then the first m digits of α are computable in time ≤ C2 22Cm for some constant C. This implies that, say, functions n π+1, n e2 and n α for all rational numbers α ≥ 4 are equivalent to the Dehn functions of some finitely presented group and that n π and n α for all rational numbers α ≥ 3 are equivalent to the smallest isodiametric functions of finitely presented groups.