Bredon cohomological dimensions for groups acting on CAT(0)-spaces
Bredon cohomological dimensions for groups acting on CAT(0)-spaces
Let $G$ be a group acting isometrically with discrete orbits on a separable complete CAT(0)-space of bounded topological dimension. Under certain conditions, we give upper bounds for the Bredon cohomological dimension of $G$ for the families of finite and virtually cyclic subgroups. As an application, we prove that the mapping class group of any closed, connected, and orientable surface of genus $g ≥ 2$ admits a $9g-8)-dimensional classifying space with virtually cyclic stabilizers. In addition, our results apply to fundamental groups of graphs of groups and groups acting on Euclidean buildings. In particular, we show that all finitely generated linear groups of positive characteristic have a finite dimensional classifying space for proper actions and a finite dimensional classifying space for the family of virtually cyclic subgroups. We also show that every generalized Baumslag–Solitar group has a 3-dimensional model for the classifying space with virtually cyclic stabilizers.
1231-1265
Degrijse, Dieter
fb33133b-fe90-42a5-8214-409c210906df
Petrosyan, Nansen
f169cfd6-aeee-4ad2-b147-0bf77dd1f9b6
2015
Degrijse, Dieter
fb33133b-fe90-42a5-8214-409c210906df
Petrosyan, Nansen
f169cfd6-aeee-4ad2-b147-0bf77dd1f9b6
Degrijse, Dieter and Petrosyan, Nansen
(2015)
Bredon cohomological dimensions for groups acting on CAT(0)-spaces.
Groups, Geometry and Dynamics, 9 (4), .
(doi:10.4171/GGD/339).
Abstract
Let $G$ be a group acting isometrically with discrete orbits on a separable complete CAT(0)-space of bounded topological dimension. Under certain conditions, we give upper bounds for the Bredon cohomological dimension of $G$ for the families of finite and virtually cyclic subgroups. As an application, we prove that the mapping class group of any closed, connected, and orientable surface of genus $g ≥ 2$ admits a $9g-8)-dimensional classifying space with virtually cyclic stabilizers. In addition, our results apply to fundamental groups of graphs of groups and groups acting on Euclidean buildings. In particular, we show that all finitely generated linear groups of positive characteristic have a finite dimensional classifying space for proper actions and a finite dimensional classifying space for the family of virtually cyclic subgroups. We also show that every generalized Baumslag–Solitar group has a 3-dimensional model for the classifying space with virtually cyclic stabilizers.
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e-pub ahead of print date: 16 November 2015
Published date: 2015
Organisations:
Pure Mathematics
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Local EPrints ID: 390768
URI: http://eprints.soton.ac.uk/id/eprint/390768
ISSN: 1661-7207
PURE UUID: 681d7cc2-3d06-449f-8e77-64090200de04
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Date deposited: 07 Apr 2016 09:22
Last modified: 15 Mar 2024 03:49
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Author:
Dieter Degrijse
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