Cohomological Finiteness Conditions in Bredon Cohomology

Details Cohomological Finiteness Conditions in Bredon Cohomology Cohomological Finiteness Conditions in Bredon Cohomology

2009-03-24

arxiv.org/abs/0903.4079v1

Mathematics

Group Theory

12 pages

Scientific paper

We show that soluble groups $G$ of type Bredon-$\FP_{\infty}$ with respect to the family of all virtually cyclic subgroups of $G$ are always virtually c\yclic. In such a group centralizers of elements are of type $\FP_{\infty}$. 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 cycl\ic 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-nilpo\tent, abelian-by-polycyclic or (nilpotent of class 2)-by-abelian groups.

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