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We construct examples of smooth 4-dimensional manifolds M supporting a locally CAT(0)-metric, whose universal cover X satisfy Hruska's isolated flats condition, and contain 2-dimensional flats F with the property that the boundary at infinity of F defines a nontrivial knot in the boundary at infinity of X. As a consequence, we obtain that the fundamental group of M cannot be isomorphic to the fundamental group of any Riemannian manifold of nonpositive sectional curvature. In particular, M is a locally CAT(0)-manifold which does not support any Riemannian metric of nonpositive sectional curvature., Comment: 20 pages, 3 figures

Kropholler's class of groups is the smallest class of groups which contains all finite groups and is closed under the following operator: whenever $G$ admits a finite-dimensional contractible $G$-CW-complex in which all stabilizer groups are in the class, then $G$ is itself in the class. Kropholler's class admits a hierarchical structure, i.e., a natural filtration indexed by the ordinals. For example, stage 0 of the hierarchy is the class of all finite groups, and stage 1 contains all groups of finite virtual cohomological dimension. We show that for each countable ordinal $\alpha$, there is a countable group that is in Kropholler's class which does not appear until the $\alpha+1$st stage of the hierarchy. Previously this was known only for $\alpha= 0$, 1 and 2. The groups that we construct contain torsion. We also review the construction of a torsion-free group that lies in the third stage of the hierarchy., Comment: 9 pages

We construct finitely generated groups with strong fixed point properties. Let $\mathcal{X}_{ac}$ be the class of Hausdorff spaces of finite covering dimension which are mod-$p$ acyclic for at least one prime $p$. We produce the first examples of infinite finitely generated groups $Q$ with the property that for any action of $Q$ on any $X\in \mathcal{X}_{ac}$, there is a global fixed point. Moreover, $Q$ may be chosen to be simple and to have Kazhdan's property (T). We construct a finitely presented infinite group $P$ that admits no non-trivial action by diffeomorphisms on any smooth manifold in $\mathcal{X}_{ac}$. In building $Q$, we exhibit new families of hyperbolic groups: for each $n\geq 1$ and each prime $p$, we construct a non-elementary hyperbolic group $G_{n,p}$ which has a generating set of size $n+2$, any proper subset of which generates a finite $p$-group., Comment: Version 2: 29 pages. This is the final published version of the article

We compute the l^2-Betti numbers of the complement of a finite collection of affine hyperplanes in complex space. At most one of the l^2-Betti numbers is non-zero., Comment: v2: Correction to proof of main theorem

Given a Coxeter system $(W,S)$ and a positive real multiparameter $\bq$, we study the "weighted $L^2$-cohomology groups," of a certain simplicial complex $\Sigma$ associated to $(W,S)$. These cohomology groups are Hilbert spaces, as well as modules over the Hecke algebra associated to $(W,S)$ and the multiparameter $q$. They have a "von Neumann dimension" with respect to the associated "Hecke - von Neumann algebra," $N_q$. The dimension of the $i^th$ cohomology group is denoted $b^i_q(\Sigma)$. It is a nonnegative real number which varies continuously with $q$. When $q$ is integral, the $b^i_q(\Sigma)$ are the usual $L^2$-Betti numbers of buildings of type $(W,S)$ and thickness $q$. For a certain range of $q$, we calculate these cohomology groups as modules over $N_q$ and obtain explicit formulas for the $b^i_q(\Sigma)$. The range of $q$ for which our calculations are valid depends on the region of convergence of the growth series of $W$. Within this range, we also prove a Decomposition Theorem for $N_q$, analogous to a theorem of L. Solomon on the decomposition of the group algebra of a finite Coxeter group., Comment: minor changes

Many examples of nonpositively curved closed manifolds arise as blow-ups of projective hyperplane arrangements. If the hyperplane arrangement is associated to a finite reflection group W, and the blow-up locus is W-invariant, then the resulting manifold M will admit a cell decomposition whose maximal cells are all combinatorially isomorphic to a given convex polytope P. In other words, M admits a tiling with tile P. The universal covers of such examples yield tilings of R^n whose symmetry groups are generated by involutions but are not, in general, reflection groups. We begin a study of these ``mock reflection groups'', and develop a theory of tilings that includes the examples coming from blow-ups and that generalizes the corresponding theory of reflection tilings. We apply our general theory to classify the examples coming from blow-ups in the case where the tile P is either the permutohedron or the associahedron.

The Baumslag-Solitar groups and their certain variations are a-T-menable. This is proved by embeding them into topological groups and studying representation theoretic properties of the latter. The paper is motivated by the questions of A. Valette., Comment: 3 pages, no figures

Coxeter groups admit amenable actions on compact spaces. Moreover, they have finite asymptotic dimension., Comment: 5 pages, AMSTeX