Cohomological invariants and the classifying space for proper actions

TL;DR

This paper explores cohomological invariants related to classifying spaces for proper group actions, addressing key open questions and providing new insights into the cohomological dimensions of specific groups.

Contribution

It reduces the problem of subadditivity of F-cohomological dimension to prime order extensions and characterizes the cohomological properties of certain branch groups.

Findings

F-cohomological dimension subadditivity problem reduced to prime order extensions

Finitely generated regular branch groups have infinite rational cohomological dimension

The first Grigorchuk group G is not in Kropholler's class HF

Abstract

We investigate two open questions in a cohomology theory relative to the family of finite subgroups. The problem of whether the F-cohomological dimension is subadditive is reduced to extensions by groups of prime order. We show that every finitely generated regular branch group has infinite rational cohomological dimension. Moreover, we prove that the first Grigorchuk group G is not contained in Kropholler's class HF .