Weak 𝒵–structures for some classes of groups

Guilbault, Craig R

doi:10.2140/agt.2014.14.1123

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

Craig R Guilbault

Algebraic & Geometric Topology 14 (2014) 1123–1152

DOI: 10.2140/agt.2014.14.1123

Abstract

Motivated by the usefulness of boundaries in the study of $δ$ –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 “ $Z$ –structure” on a group $G$ . These $Z$ –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 $Z$ –structures; aside from the (easy) fact that any such $G$ must have type F, ie, $G$ 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 general existence theorems for weak $Z$ –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 $Z$ –structure.

Theorem B If $G$ admits a finite K $(G, 1)$ complex $K$ such that the $G$ –action on $\tilde{K}$ contains $1 \neq j \in G$ properly homotopic to ${id}_{\tilde{K}}$ , 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.

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

Keywords

$Z$–set, $Z$–compactification, $Z$–structure, $Z$–boundary, weak $Z$–structure, weak $Z$–boundary, group extension, approximate fibration

Mathematical Subject Classification 2010

Primary: 57M07, 20F65

Secondary: 57N20

References

Publication

Received: 23 August 2013

Accepted: 2 September 2013

Published: 21 March 2014

Authors

Craig R Guilbault
Department of Mathematical Sciences University of Wisconsin-Milwaukee PO Box 413 Milwaukee, WI 53201 USA

Volume 14, issue 2 (2014)