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

Craig R Guilbault

Algebraic & Geometric Topology 14 (2014) 1123–1152

Motivated by the usefulness of boundaries in the study of ή–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 K̃ contains 1≠j ∈ G properly homotopic to idK̃, 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 × â„€ admits a weak Z–structure. As another corollary of Theorem B, every type F group with a nontrivial center admits a weak Z–structure.

$Z$–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
Received: 23 August 2013
Accepted: 2 September 2013
Published: 21 March 2014
Craig R Guilbault
Department of Mathematical Sciences
University of Wisconsin-Milwaukee
PO Box 413
Milwaukee, WI 53201