Volume 14, issue 2 (2014)

Download this article
Download this article For screen
For printing
Recent Issues

Volume 24
Issue 6, 2971–3570
Issue 5, 2389–2970
Issue 4, 1809–2387
Issue 3, 1225–1808
Issue 2, 595–1223
Issue 1, 1–594

Volume 23, 9 issues

Volume 22, 8 issues

Volume 21, 7 issues

Volume 20, 7 issues

Volume 19, 7 issues

Volume 18, 7 issues

Volume 17, 6 issues

Volume 16, 6 issues

Volume 15, 6 issues

Volume 14, 6 issues

Volume 13, 6 issues

Volume 12, 4 issues

Volume 11, 5 issues

Volume 10, 4 issues

Volume 9, 4 issues

Volume 8, 4 issues

Volume 7, 4 issues

Volume 6, 5 issues

Volume 5, 4 issues

Volume 4, 2 issues

Volume 3, 2 issues

Volume 2, 2 issues

Volume 1, 2 issues

The Journal
About the Journal
Editorial Board
Subscriptions
 
Submission Guidelines
Submission Page
Policies for Authors
Ethics Statement
 
ISSN 1472-2739 (online)
ISSN 1472-2747 (print)
Author Index
To Appear
 
Other MSP Journals
Weak $\mathcal{Z}$–structures for some classes of groups

Craig R Guilbault

Algebraic & Geometric Topology 14 (2014) 1123–1152
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 K̃ contains 1j 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 × 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