Bredon cohomological dimensions for groups acting on CAT(0)-spaces

Dieter Dries Degrijse; Nansen  Petrosyan

doi:10.4171/ggd/339

Bredon cohomological dimensions for groups acting on CAT(0)-spaces

Dieter Dries Degrijse, Nansen Petrosyan

Institut for Matematiske Fag

10 Citationer (Scopus)

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.

Originalsprog	Engelsk
Tidsskrift	Groups, Geometry, and Dynamics
Vol/bind	9
Udgave nummer	4
Sider (fra-til)	1231–1265
ISSN	1661-7207
DOI	https://doi.org/10.4171/ggd/339
Status	Udgivet - 2015

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10.4171/ggd/339

http://www.ems-ph.org/journals/show_pdf.php?issn=1661-7207&vol=9&iss=4&rank=7

Citationsformater

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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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T1 - Bredon cohomological dimensions for groups acting on CAT(0)-spaces

AU - Degrijse, Dieter Dries

AU - Petrosyan, Nansen

PY - 2015

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N2 - 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.

AB - 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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DO - 10.4171/ggd/339

M3 - Journal article

SN - 1661-7207

VL - 9

SP - 1231

EP - 1265

JO - Groups, Geometry, and Dynamics

JF - Groups, Geometry, and Dynamics

IS - 4

ER -

Bredon cohomological dimensions for groups acting on CAT(0)-spaces

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