Cohomological Finiteness Conditions in Bredon Cohomology

Abstract

We show that any soluble group $G$ of type Bredon-$\FP_{\infty}$ with respect to the family of all virtually cyclic subgroups such that centralizers of infinite order elements are of type $\FP_{\infty}$ must be virtually cyclic. To prove this, we first reduce the problem to the case of polycyclic groups and then we show 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 $n\le 3$ and restrict to abelian-by-nilpotent, abelian-by-polycyclic or (nilpotent of class 2)-by-abelian groups.

Corrected a mistake in Lemma 2.4 of the previous version, which had an effect on the results in Section 5 (the condition that all centralisers of infinite order elements are of type $F{P}_{\infty }$ was added)

References in corpus (2)

• On the classifying space of the family of virtually cyclic subgroups
• On the finiteness of the classifying space for the family of virtually cyclic subgroups

Cited by in corpus (4)

• Equivariant coarse homotopy theory and coarse algebraic $\mathbf{K}$-homology
• Brown's criterion in Bredon homology
• Some results related to finiteness properties of groups for families of subgroups
• Complete Bredon cohomology and its applications to hierarchically defined groups