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 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 × 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
Guilbault, Craig R 1
@article{10_2140_agt_2014_14_1123,
author = {Guilbault, Craig R},
title = {Weak {\ensuremath{\mathscr{Z}}{\textendash}structures} for some classes of groups},
journal = {Algebraic and Geometric Topology},
pages = {1123--1152},
year = {2014},
volume = {14},
number = {2},
doi = {10.2140/agt.2014.14.1123},
url = {http://geodesic.mathdoc.fr/articles/10.2140/agt.2014.14.1123/}
}
Guilbault, Craig R. Weak 𝒵–structures for some classes of groups. Algebraic and Geometric Topology, Tome 14 (2014) no. 2, pp. 1123-1152. doi: 10.2140/agt.2014.14.1123
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