### Abstract

Let G be an infinite discrete group. A classifying space for proper actions of G is a proper G-CW-complex X such that the fixed point sets X^H are contractible for all finite subgroups H of G. In this paper we consider the stable analogue of the classifying space for proper actions in the category of proper G-spectra and study its finiteness properties. We investigate when G admits a stable classifying space for proper actions that is finite or of finite type and relate these conditions to the compactness of the sphere spectrum in the homotopy category of proper G-spectra and to classical finiteness properties of the Weyl groups of finite subgroups of G. Finally, if the group G is virtually torsion-free we also show that the smallest possible dimension of a stable classifying space for proper actions coincides with the virtual cohomological dimension of G, thus providing the first geometric interpretation of the virtual cohomological dimension of a group.

Original language | English |
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Pages (from-to) | 1169-1196 |

Number of pages | 28 |

Journal | Journal of Topology |

Volume | 10 |

Issue number | 4 |

Early online date | 8 Dec 2017 |

DOIs | |

Publication status | Published - Dec 2017 |

### Keywords

- 20J05
- 55P42
- 55P91 (primary)

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## Profiles

### Irakli Patchkoria

- School of Natural & Computing Sciences, Mathematical Science - Lecturer
- Mathematical Sciences (Research Theme)

Person: Academic

## Cite this

Bárcenas, N., Degrijse, D., & Patchkoria, I. (2017). Stable finiteness properties of infinite discrete groups.

*Journal of Topology*,*10*(4), 1169-1196. https://doi.org/10.1112/topo.12035