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Cohomological finiteness conditions in Bredon cohomology

Cohomological finiteness conditions in Bredon cohomology
Cohomological finiteness conditions in Bredon cohomology
We show that soluble groups G of type Bredon-FP? 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?. 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.
0024-6093
124-136
Kochloukova, D.H.
3a54fa65-f950-44b6-beb3-28030dfb2558
Martinez-Perez, C.
cb55fc61-2a74-4072-93d3-74a882ad5fdc
Nucinkis, B.E.A.
41b6a278-1866-4585-84f5-601bb85636a2
Kochloukova, D.H.
3a54fa65-f950-44b6-beb3-28030dfb2558
Martinez-Perez, C.
cb55fc61-2a74-4072-93d3-74a882ad5fdc
Nucinkis, B.E.A.
41b6a278-1866-4585-84f5-601bb85636a2

Kochloukova, D.H., Martinez-Perez, C. and Nucinkis, B.E.A. (2009) Cohomological finiteness conditions in Bredon cohomology. Bulletin of the London Mathematical Society, 43 (1), 124-136. (doi:10.1112/blms/bdq088).

Record type: Article

Abstract

We show that soluble groups G of type Bredon-FP? 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?. 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.

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More information

Published date: 24 March 2009
Additional Information: arXiv:0903.4079v1

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Local EPrints ID: 63092
URI: http://eprints.soton.ac.uk/id/eprint/63092
ISSN: 0024-6093
PURE UUID: 70f4d05e-ebb2-477a-b1f6-18841384b0c6

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Date deposited: 11 Sep 2008
Last modified: 15 Mar 2024 11:34

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Contributors

Author: D.H. Kochloukova
Author: C. Martinez-Perez
Author: B.E.A. Nucinkis

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