Weak $\mathcal{Z}$–structures for some classes of groups

Abstract

Motivated by the usefulness of boundaries in the study of $\delta$–hyperbolic and CAT(0) groups, Bestvina introduced a general axiomatic approach to group boundaries, with a goal of extending the theory and application of boundaries to larger classes of groups. The key definition is that of a “$\mathcal{Z}$–structure” on a group $G$. These $\mathcal{Z}$–structures, along with several variations, have been studied and existence results have been obtained for a variety of new classes of groups. Still, relatively little is known about the general question of which groups admit any of the various $\mathcal{Z}$–structures; aside from the (easy) fact that any such $G$ must have type F, ie, $G$ must admit a finite K$\left(G,1\right)$. In fact, Bestvina has asked whether every type F group admits a $\mathcal{Z}$–structure or at least a “weak” $\mathcal{Z}$–structure.

In this paper we prove some general existence theorems for weak $\mathcal{Z}$–structures. The main results are as follows.

Theorem A If $G$ is an extension of a nontrivial type F group by a nontrivial type F group, then $G$ admits a weak $\mathcal{Z}$–structure.

Theorem B If $G$ admits a finite K$\left(G,1\right)$ complex $K$ such that the $G$–action on $\tilde{K}$ contains $1\ne j\in G$ properly homotopic to ${id}_{\tilde{K}}$, then $G$ admits a weak $\mathcal{Z}$–structure.

Theorem C If $G$ has type F and is simply connected at infinity, then $G$ admits a weak $\mathcal{Z}$–structure.

As a corollary of Theorem A or B, every type F group admits a weak $\mathcal{Z}$–structure “after stabilization”; more precisely: if $H$ has type F, then $H\times \mathbb{Z}$ admits a weak $\mathcal{Z}$–structure. As another corollary of Theorem B, every type F group with a nontrivial center admits a weak $\mathcal{Z}$–structure.