Some groups of type \(VF\) (Q1401465)

A group is of type \(\mathbf F\) if it has a finite classifying space. A group is of type \(\mathbf{VF}\) if it has a finite index subgroup of type \(\mathbf F\). \textit{M. Bestvina} and \textit{N. Brady} [Invent. Math. 129, No. 3, 445-470 (1997; Zbl 0888.20021)] give a method for constructing torsion-free groups with prescribed homological finiteness properties. Using some of the methods of this paper, the authors produce, among many other, examples of groups of type \(\mathbf{VF}\) satisfying that the centralizers of some elements of finite order are not of type \(\mathbf{VF}\), groups of type \(\mathbf{VF}\) that contain infinitely many conjugacy classes of finite subgroups. Moreover, they also provide groups for which the minimal dimension of a universal proper classifying space is strictly larger than the virtual cohomological dimension of the group. These examples show that some properties are not preserved under taking finite extensions of groups. The main result may read as follows: given a finite flag complex \(L\) and a group \(Q\) of automorphisms of \(L\), they produce a group \(\widetilde H=H_L\rtimes Q\), where \(H_L\) is provided by the Bestiva-Brady machinery, and such that the homological properties of \(\widetilde H\) are determined by the homotopy type of \(L\) and the \(P\)-fixed points in \(L\) for each \(P\leq Q\).