On the dimension of groups that satisfy certain conditions on their finite subgroups

TL;DR

This paper proves that for certain groups with specific properties, the Bredon cohomological dimension equals the virtual cohomological dimension, including examples like Fuchsian and 3-manifold groups.

Contribution

It establishes the equality of two important cohomological dimensions for a broad class of groups satisfying properties (M) and (NM).

Findings

Bredon cohomological and virtual cohomological dimensions coincide under given conditions.

Includes examples such as cocompact Fuchsian, one relator, Hilbert modular, and 3-manifold groups.

Provides a unifying result for these classes of groups.

Abstract

We prove that the Bredon cohomological dimension and the virtual cohomological dimension coincide for groups that admit a cocompact model for $\underline{E}G$ and satisfy properties (M) and (NM). Among the examples of groups satisfying these hypothesis are cocompact and arithmetic Fuchsian groups, one relator groups, the Hilbert modular group and $3$-manifold groups.