Stable finiteness properties of infinite discrete groups. (English) Zbl 1485.55014
Summary: 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.
MSC:
| 55P91 | Equivariant homotopy theory in algebraic topology |
| 20J05 | Homological methods in group theory |
| 55P42 | Stable homotopy theory, spectra |
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