Weak Z-structures for some classes of groups

作者：Guilbault Craig R^*

来源：Algebraic and Geometric Topology, 2014, 14(2): 1123-1152.

DOI：10.2140/agt.2014.14.1123

Motivated by the usefulness of boundaries in the study of delta-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 (K) over tilde contains 1 not equal j is an element of G properly homotopic to id((K) over tilde), 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 x 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.

• 出版日期2014