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.