Abstract
Motivated by the usefulness of boundaries in the study of –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 “–structure” on a group . These –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 –structures; aside from the (easy) fact that any such must have type F, ie, must admit a finite K. In fact, Bestvina has asked whether every type F group admits a –structure or at least a “weak” –structure.
In this paper we prove some general existence theorems for weak –structures. The main results are as follows.
Theorem A If is an extension of a nontrivial type F group by a nontrivial type F group, then admits a weak –structure.
Theorem B If admits a finite K complex such that the –action on contains properly homotopic to , then admits a weak –structure.
Theorem C If has type F and is simply connected at infinity, then admits a weak –structure.
As a corollary of Theorem A or B, every type F group admits a weak –structure “after stabilization”; more precisely: if has type F, then admits a weak –structure. As another corollary of Theorem B, every type F group with a nontrivial center admits a weak –structure.
Citation
Craig R Guilbault. "Weak $\mathcal{Z}$–structures for some classes of groups." Algebr. Geom. Topol. 14 (2) 1123 - 1152, 2014. https://doi.org/10.2140/agt.2014.14.1123
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