Cohomological Finiteness Conditions in Bredon Cohomology

TL;DR

This paper proves that soluble groups with certain Bredon finiteness properties related to virtually cyclic subgroups are virtually cyclic, using reductions to polycyclic groups and analyzing conjugacy classes.

Contribution

It establishes that soluble groups of type Bredon-$ ext{FP}_ ext{infinity}$ with specific conditions are necessarily virtually cyclic, extending understanding of cohomological finiteness in group theory.

Findings

Soluble groups of type Bredon-$ ext{FP}_ ext{infinity}$ are virtually cyclic under certain conditions.

Polycyclic-by-finite groups with finitely many conjugacy classes of maximal virtually cyclic subgroups are virtually cyclic.

Refinements show similar results for groups with weaker finiteness conditions and specific structural restrictions.

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 $n \leq 3$ and restrict to abelian-by-nilpotent, abelian-by-polycyclic or (nilpotent of class 2)-by-abelian groups.