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Cohomological finiteness conditions in Bredon cohomology

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.