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, thenadmits a weak–structure.
Theorem B If admits a finite Kcomplex suchthat the –actionon containsproperlyhomotopic to ,then admits aweak –structure.
Theorem C If has type F and is simply connected at infinity, thenadmits 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.