We introduce a family of conditions on a simplicial complex that we call local k-largeness (k ≥ 6 is an integer). They are simply stated, combinatorial and easily checkable. One of our themes is that local 6-largeness is a good analogue of the non-positive curvature: locally 6-large spaces have many properties similar to non-positively curved ones. However, local 6-largeness neither implies nor is implied by non-positive curvature of the standard metric. One can think of these results as a higher dimensional version of small cancellation theory. On the other hand, we show that k-largeness implies non-positive curvature if k is sufficiently large. We also show that locally k-large spaces exist in every dimension. We use this to answer questions raised by D. Burago, M. Gromov and I. Leary.

We present an update of Garland's work on the cohomology of certain groups, construct a class of groups many of which satisfy Kazhdan's Property (T) and show that properly discontinuous and cocompact groups of automorphisms of (4, 4) or (6, 3)-complexes do not satisfy Property (T).For a simplicial complex X, the link X v of X at a vertex v of X is defined to be the subcomplex consisting of all simplices τ of X which do not contain v, but whose union with v is a simplex of X. If dim X = 2, then X v is a graph for any vertex v of X.For a finite graph L with set of vertices V L , consider the Laplacian ∆ on the space of real valued functions on V L defined bywhere Af (v) is the mean value of f on the vertices adjacent to v. Clearly, ∆ is a self adjoint operator. We denote by κ = κ(L) its smallest positive eigenvalue. Theorem 1. Let X be a locally finite 2-dimensional simplicial complex such that(1) for any vertex v of X the link X v is connected;(2) there is an ε > 0 such that κ(X v ) > 1 2 + ε for each vertex v of X. Let Γ be a properly discontinuous group of automorphisms of X and ρ be a unitary representation of Γ. Then L 2 H 1 (X, ρ) = 0, where L 2 H(X, ρ) denotes the cohomology of the complex of mod Γ square integrable cochains on X which are twisted by ρ.

Abstract.We construct examples of Gromov hyperbolic Coxeter groups of arbitrarily large dimension. We also extend Vinberg's theorem to show that if a Gromov hyperbolic Coxeter group is a virtual Poincaré duality group of dimension n, then n ≤ 61.Coxeter groups acting on their associated complexes have been extremely useful source of examples and insight into nonpositively curved spaces over last several years. Negatively curved (or Gromov hyperbolic) Coxeter groups were much more elusive. In particular their existence in high dimensions was in doubt.In 1987 Gabor Moussong [M] conjectured that there is a universal bound on the virtual cohomological dimension of any Gromov hyperbolic Coxeter group. This question was also raised by Misha Gromov [G] (who thought that perhaps any construction of high dimensional negatively curved spaces requires nontrivial number theory in the guise of arithmetic groups in an essential way), and by Mladen Bestvina [B2].In the present paper we show that high dimensional Gromov hyperbolic Coxeter groups do exist, and we construct them by geometric or group theoretic but not arithmetic means. [M] that this theorem reduces to the existence of a simplicial complex L n which is (i) flag, (ii) contains no empty square and (iii) satisfies certain homological condition (which is implied by say requiring that L n is an oriented pseudomanifold of dimension n−1). Such a complex gives rise to the Coxeter group (W, S) with generators indexed by vertices of L n and relations (x i x j ) 2 = 1 iff vertices i, j are connected by an edge. Then (i) implies that the nerve of (W, S) is L n , (ii) implies hyperbolicity and (iii) implies that vcd(W ) = n. We construct simplicial complexes required for the proof of Theorem 1 using complexes of groups technique of [BH]. The resulting spaces are interesting on their own right and we study them further in a forthcoming paper. Let us mention here only that the L n we construct turn out to be K(π, 1) spaces. Mathematics Subject Classification (2000Gromov The right angled case of Vinberg theorem has the following corollary dealing with arbitrary cubical complexes. Corollary. If M is a manifold of dimension greater than 4, then it does not admit a negatively curved piecewise hyperbolic cubical metric.The content of the paper is as follows. Section 1 contains a necessary background on Coxeter groups. Section 2 deals with the nonexistence results. We prove Theorem 2 following closely Vinberg's paper, discussing only points which are specific to our situation. We do not claim much originality here. The reader should be ready to consult [V]. We also establish a slightly stronger version of the Corollary.In Section 3 we recall the simplex of groups construction in the form we need. We focus on the properties of (multi-) simplicial complexes arising as developments (coverings) of simplices of groups.In Section 4 we define (and establish properties of) special complexes of groups which we call retractible. This is done by enriching the structure of a complex of groups...

Abstract. Let Γ(n, p) denote the binomial model of a random triangular group. We show that there exist constants c, C > 0 such that if p ≤ c/n 2 , then a.a.s. Γ(n, p) is free and if p ≥ Clog n/n 2

727Filling invariants of systolic complexes and groups TADEUSZ JANUSZKIEWICZ JACEKŚWIĄTKOWSKISystolic complexes are simplicial analogues of nonpositively curved spaces. Their theory seems to be largely parallel to that of CAT(0) cubical complexes.We study the filling radius of spherical cycles in systolic complexes, and obtain several corollaries. We show that a systolic group can not contain the fundamental group of a nonpositively curved Riemannian manifold of dimension strictly greater than 2, although there exist word hyperbolic systolic groups of arbitrary cohomological dimension.We show that if a systolic group splits as a direct product, then both factors are virtually free. We also show that systolic groups satisfy linear isoperimetric inequality in dimension 2.