Some groups of type VF

Ian J. Leary and Brita E. A. Nucinkis

Abstract.
A group is of type VF if it has 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.  It 
follows that a group $G$ of type VF need not admit a finite-type 
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 $GL_m(\zz)$ for sufficiently
large $m$.  Some applications to $K$-theory are also considered.