Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)We show that soluble groups G of type Bredon-FP(infinity) 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(infinity). 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-FP(n) for some n < 3 and restrict to abelian-by-nilpotent, abelian-by-polycyclic or (nilpotent of class 2)-by-abelian groups.431124136EPSRC [EP/F045395/1]LMS [4708]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)Gobierno de Aragon[MTM2007-68010-C03-01]Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)EPSRC [EP/F045395/1]LMS [4708][MTM2007-68010-C03-01]