Weak Z-structures for some classes of groups

TL;DR

This paper investigates conditions under which groups admit weak Z-structures, providing general theorems that expand understanding of boundary constructions in geometric group theory.

Contribution

The paper proves new theorems establishing the existence of weak Z-structures for broad classes of groups, including extensions and groups with specific topological properties.

Findings

Groups that are extensions of type F groups admit Z-structures.

Groups with certain actions on universal covers admit weak Z-structures.

Type F groups that are simply connected at infinity admit weak Z-structures.

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.