Weak 𝒵–structures for some classes of groups

Abstract

Motivated by the usefulness of boundaries in the study of hyperbolic and CAT(0) groups, Bestvina introduced a general approach to group boundaries via the notion of a Z-structure on a group G. Several variations on Z-structures have been studied and existence results have been obtained for some very specific classes of groups. However, little is known about the general question of which groups admit any of the various Z-structures, aside from the (easy) fact that any such G must have type F, i.e., G must admit a finite K(G,1). In fact, Bestvina has asked whether every type F group admits a Z-structure or at least a "weak" Z-structure. In this paper we prove some rather general existence theorems for weak Z-structures. Among our results are the following: Theorem A. If G is an extension of a nontrivial type F group by a nontrivial type F group, then G admits a Z-structure. Theorem B. If G admits a finite K(G,1) complex K such that the corresponding G-action on the universal cover contains a non-identity element properly homotopic to the identity, then G admits a weak Z-structure. Theorem C. If G has type F and is simply connected at infinity, then G admits a weak Z-structure.