Cohomological finiteness conditions in Bredon cohomology

Abstract

We show that soluble groups G of type Bredon-FP∞ with respect to the family of all virtually cyclic subgroups of G are always virtually cyclic. In such a group centralizers of elements are of type FP∞. We show that this implies that the group is polycyclic. Another important ingredient of the proof is that a polycyclic-by-finite group with finitely many conjugacy classes of maximal virtually cyclic subgroups is virtually cyclic. Finally we discuss refinements of this result: we only impose the property Bredon-FPn for some n ⩽ 3 and restrict to abelian-by-nilpotent, abelian-by-polycyclic or (nilpotent of class 2)-by-abelian groups.