On dimensions of groups with cocompact classifying spaces for proper actions

TL;DR

This paper constructs new groups with minimal possible dimensions for their classifying spaces for proper actions, showing these groups can have cocompact models even with low cohomological dimensions.

Contribution

It introduces the first examples of groups with these properties, including groups with equal virtual and Bredon cohomological dimensions that lack 2-dimensional models.

Findings

Groups with virtual cohomological dimension less than the minimal model dimension.

Existence of groups with equal virtual and Bredon cohomological dimensions that do not admit 2-dimensional models.

Abstract

We construct groups G that are virtually torsion-free and have virtual cohomological dimension strictly less than the minimal dimension for any model for the classifying space for proper actions of G. They are the first examples that have these properties and also admit cocompact models for this classifying space. We exhibit groups G whose virtual cohomological dimension and Bredon cohomological dimension are two that do not admit any 2-dimensional contractible proper G-CW-complex.