  Motivated by the usefulness of boundaries in the study of hyperbolic and
CAT(0) groups, Bestvina introduced a general approach to group boundaries via
the notion of a Z-structure on a group G. Several variations on Z-structures
have been studied and existence results have been obtained for some very
specific classes of groups. However, 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, i.e., 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 rather general
existence theorems for weak Z-structures. Among our results are the following:
  Theorem A. If G is an extension of a nontrivial type F group by a nontrivial
type F group, then G admits a Z-structure.
  Theorem B. If G admits a finite K(G,1) complex K such that the corresponding
G-action on the universal cover contains a non-identity element properly
homotopic to the identity, 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.
