### Some groups of type *VF*

### I. J. Leary and Brita E. A. Nucinkis

A group *G* is of type *VF* if it contains a finite-index
subgroup which has a finite classifying space. We construct groups of
type *VF* in which the centralizers of some elements of finite
order are not of type *VF* and groups of type *VF*
containing infinitely many conjugacy classes of finite subgroups.
(Contrast this with a result of K. S. Brown that implies that groups
of type *VF* contain finitely many conjugacy classes of subgroups
of prime power order.)

From these examples it follows that a group *G* of type *VF*
need not admit a finite-type or finite classifying space for proper
actions (sometimes also called the universal proper *G*-space).

We construct groups *G* for which the minimal dimension of a
universal proper *G*-space is strictly greater than the virtual
cohomological dimension of *G*. Each of our groups embeds in a
general linear group over the rational integers. Applications to
algebraic K-theory of group algebras and topological K-theory of
group C*-algebras are also considered.

The groups are constructed as finite extensions of Bestvina-Brady
groups.

For some reason, the Mathematical Review of this paper mentions only
the construction of groups of type VF containing infinitely many
conjugacy classes of finite subgroups, and makes no mention of any
of the other results in the paper.