Stable finiteness properties of infinite discrete groups

TL;DR

This paper explores the stable analogue of classifying spaces for proper actions of infinite discrete groups, linking their finiteness properties to homotopy-theoretic and algebraic invariants, and providing a geometric interpretation of virtual cohomological dimension.

Contribution

It introduces the concept of stable classifying spaces for proper actions, analyzes their finiteness properties, and relates these to classical invariants like the virtual cohomological dimension.

Findings

Stable classifying spaces can be finite or of finite type under certain conditions.

The smallest dimension of a stable classifying space equals the virtual cohomological dimension for virtually torsion-free groups.

Provides a geometric interpretation of virtual cohomological dimension.

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.