Abstract. We present a general condition, based on the idea of n-generating subgroup sets, which implies that a given character χ ∈ Hom(G, ) represents a point in the homotopical or homological Σ-invariants of the group G. Let G be a finite simplicial graph, G the flag complex induced by G, and GG the graph group, or 'right angled Artin group', defined by G. We use our result on n-generating subgroup sets to describe the homotopical and homological Σ-invariants of GG in terms of the topology of subcomplexes of G. In particular, this work determines the finiteness properties of kernels of maps from graph groups to abelian groups. This is the first complete computation of the Σ-invariants for a family of groups whose higher invariants are not determined -either implicitly or explicitly -by Σ 1 . Mathematics Subject Classification (1991). 57M07, 20F36.

Recent work of Gromov, Epstein, Cannon, Thurston and many others has generated strong interest in the geometric and algorithmic structure of finitely generated infinite groups. (See [16],[17] and [14].) Many of these structures are preserved by taking graph products. The graph product of groups (not to be confused with the fundamental group of a graph of groups) is a product mixing direct and free products. Whether the product between two groups in the graph product is free or direct is determined by a simplicial graph. Given a simplicial graph we say that two vertices are adjacent if they are joined by a single edge. Definition. Given a finite simplicial graph G with a group (or monoid) attached to each vertex, the associated graph product is the group (monoid) generated by each of the vertex groups (monoids) with the added relations that elements of distinct adjacent vertex groups commute. Graph products were defined by Green [15], and have also been studied by Chiswell [8], [9], [10]. Graph products are a generalization of "semifree groups"

Abstract. A Garside group is a group admitting a finite lattice generating set D. Using techniques developed by Bestvina for Artin groups of finite type, we construct K(π, 1)s for Garside groups. This construction shows that the (co)homology of any Garside group G is easily computed given the lattice D, and there is a simple sufficient condition that implies G is a duality group. The universal covers of these K(π, 1)s enjoy Bestvina's weak non-positive curvature condition. Under a certain tameness condition, this implies that every solvable subgroup of G is virtually abelian.

Given a finite simplicial graph g, the graph group Gg is the group with generators in one‐to‐one correspondence with the verticles of g and wuth relations stating that two generators commute if their associated vertices are adjacent in g. The Bieri–Neumann–Strebel invariant can be explicitly described in terms of the original graph g and hence there is an explicit description of the distribution of finitely generated normal subgroups of Gg with abelian quotient. We construct Eilenberg‐MacLane spaces for graph groups and find partial extensions of this work to the higher‐dimensional invariants.

Let PΣ n denote the group that can be thought of either as the group of motions of the trivial n-component link or the group of symmetric automorphisms of a free group of rank n. The integral cohomology ring of PΣ n is determined, establishing a conjecture of Brownstein and Lee. 20J06; 57M07