Dimension invariants for groups admitting a cocompact model for proper actions

Abstract

Let $G$ be a group that admits a cocompact classifying space for proper actions $X$. We derive a formula for the Bredon cohomological dimension for proper actions of $G$ in terms of the relative cohomology with compact support of certain pairs of subcomplexes of $X$. We use this formula to compute the Bredon cohomological dimension for proper actions of fundamental groups of non-positively curved simple complexes of finite groups. As an application we show that if a virtually torsion-free group acts properly and chamber transitively on a building, its virtual cohomological dimension coincides with its Bredon cohomological dimension. This covers the case of Coxeter groups and graph products of finite groups.

results on relative cohomological dimension added, 18 pages

References in corpus (3)

• The proper geometric dimension of the mapping class group
• On the Gorenstein and $\mathfrak{F}$-cohomological dimensions
• Bredon cohomological dimensions for proper actions and Mackey functors

Cited by in corpus (1)

• Stable finiteness properties of infinite discrete groups