6 research outputs found
Cohomological Finiteness Conditions in Bredon Cohomology
We show that any soluble group 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 and restrict to abelian-by-nilpotent, abelian-by-polycyclic
or (nilpotent of class 2)-by-abelian groups.Comment: 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 was added
Cohomoloical finiteness conditions in Bredon cohomology, preprint 2008
Abstract. We show that soluble groups G of type Bredon-FP â 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 FPâ. 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-FPn for some n †3 and restrict to abelian-by-nilpotent, abelian-by-polycyclic or (nilpotent of class 2)-by-abelian groups. 1