### Cohomological finiteness conditions in Bredon cohomology

### Dessislava H. Kochloukova, Conchita Martinez-Perez and Brita E. A. Nucinkis

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 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-FP_{n} for some
n less or equal 3 and restrict to abelian-by-nilpotent, abelian-by-polycyclic or (nilpotent
of class 2)-by-abelian groups.

Preprint (2008)