Abstract. We explain some simple methods to establish the property of Rapid Decay for a number of groups arising geometrically. We also give new examples of groups with the property of Rapid Decay. In particular we establish the property of Rapid Decay for all lattices in rank one Lie groups.

We show that groups satisfying Kazhdan's property (T) have no unbounded actions on finite dimensional CAT(0) cube complexes, and deduce that there is a locally CAT( \Gamma1 ) Riemannian manifold which is not homotopy equivalent to any finite dimensional, locally CAT(0) cube complex. AMS Classification numbers Primary: 20F32 Secondary: 20E42, 20G20 Keywords: Kazhdan's property (T), Tits' buildings, hyperbolic geometry, CAT(0) cube complexes, locally CAT( \Gamma1 ) spaces, Sp(n; 1)--manifolds Proposed: Walter Neumann Received: 28 October 1996 Seconded: David Gabai, Robion Kirby Accepted: 6 February 1997 1 ISSN 1364-0380 Copyright Geometry and Topology 1 1 Introduction The CAT() inequality gives a measure of the curvature of a geodesic metric space X by comparing the width of the geodesic triangles in X with those of the corresponding triangles in the simply connected Riemannian manifold of constant curvature . The theory of CAT() metric spaces is described in [2]. A geodesic met...

Abstract. We show that the Hilbert space compression of any finite dimensional CAT(0) cube complex is 1 and deduce that any discrete group acting properly, co-compactly on a CAT(0) cube complex is exact. The class of groups covered by this theorem includes free groups, finitely generated Coxeter groups, finitely generated right angled Artin groups, finitely presented groups satisfying the B(4)-T(4) small cancellation condition and all those wordhyperbolic groups satisfying the B(6) condition. Another family of examples is provided by certain canonical surgeries defined by link diagrams.

Abstract We describe a correspondence between spaces with walls and CAT(0) cube complexes. AMS Classi cation 20F65; 20E42 Keywords Space with walls, Median graph, CAT(0) cube complex

Abstract. We prove that finite index subgroups of right angled Artin groups are conjugacy separable. We then apply this result to establish various properties of other classes of groups. In particular, we show that any word hyperbolic Coxeter group contains a conjugacy separable subgroup of finite index and has a residually finite outer automorphism group. Another consequence of the main result is that Bestvina-Brady groups are

In this work we introduce a new combinatorial notion of boundary ℜC of an ω-dimensional cubing C. ℜC is defined to be the set of almost-equality classes of ultrafilters on the standard system of halfspaces of C, endowed with an order relation reflecting the interaction between the Tychonoff closures of the classes. When C arises as the dual of a cubulation – or discrete system of halfspaces – H of a CAT(0) space X (for example, the Niblo-Reeves cubulation of the Davis-Moussong complex of a finite rank Coxeter group), we show how H induces a function ρ: ∂∞X → ℜC. We develop a notion of uniformness for H, generalizing the parallel walls property enjoyed by Coxeter groups, and show that, if the pair (X, H) admits a geometric action by a group G, then the fibers of ρ form a stratification of ∂∞X graded by the order structure of ℜC. We also show how this structure computes the components of the Tits boundary of X. Finally, using our result from another paper, that the uniformness of a cubulation as above implies the local finiteness of C, we give a condition for the cocompactness of the action of G on C in terms of ρ, generalizing a result of Williams, previously known only for Coxeter groups. 1

Abstract. We prove that random groups at density less than 1 6 act freely and cocompactly on CAT(0) cube complexes, and that random groups at density less than 1 have codimension-1 subgroups. In partic-5 ular, Property (T) fails to hold at density less than 1

(MAP) if the identity map id: A → A can be approximated in the point-norm topology by a net of finite rank contractions. The main purpose of this note is to prove the following theorem. Theorem. Let Γ be a discrete group satisfying the rapid decay property with respect to a length function ℓ which is conditionally negative. Then the reduced C∗-algebra C ∗ r (Γ) has the metric approximation property. The central point of our proof is an observation that the proof of the same property for free groups due to Haagerup [2] transfers directly to this more general situation. We also note that under the same hypotheses, the Fourier algebra A(Γ) has a bounded approximate identity, which implies that it too has the MAP. A discrete group Γ satisfies property (RD) (Rapid Decay) with respect to a length function ℓ on Γ if the operator norm of any element of the group ring can be uniformly majorised by a Sobolev norm determined by ℓ. In detail, this means the following. The left action of a group Γ on itself extends to the convolution action of the group ring CΓ on the Hilbert space ℓ2 (Γ). This is the left regular representation λ of Γ which embeds the group ring in the C∗-algebra B(ℓ2 (Γ)) of all bounded linear operators on ℓ2 (Γ). The reduced C∗-algebra C ∗ r (Γ) is the C∗-subalgebra of B(ℓ2 (Γ)) generated by λ(CΓ). For any positive real number s we define a Sobolev norm associated with the length function ℓ by

We study global fixed points for actions of Coxeter groups on nonpositively curved singular spaces. In particular, we consider property FAn, an analogue of Serre’s property FA for CAT(0) complexes of higher dimension. FAn has applications to irreducible representations and to complex of groups decompositions. In this paper, we give a specific condition on Coxeter presentations that implies FAn and show that this condition is in fact equivalent to FAn for n = 1 and 2. 1

The Wythoff construction takes a d-dimensional polytope P, a subset S of {0,..., d} and returns another d-dimensional polytope P (S). If P is a regular polytope, then P (S) is vertex-transitive. This construction builds a large part of the Archimedean polytopes and tilings in dimension 3 and 4. We want to determine, which of those Wythoffians P (S) with regular P have their skeleton or dual skeleton isometrically embeddable into the hypercubes Hm and half-cubes 1 2 Hm. We find six infinite series, which, we conjecture, cover all cases for dimension d> 5 and some sporadic cases in dimension 3 and 4 (see Tables 1 and 2). Three out of those six infinite series are explained by a general result about the embedding of Wythoff construction for Coxeter groups. In the last section, we consider the Euclidean case; also, zonotopality of embeddable P (S) are addressed throughout the text.