Cohomological invariants and the classifying space for proper actions

We investigate two open questions in a cohomology theory relative to the family of finite subgroups. The problem of whether the $\mathbb{F}$-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 $\mathfrak{G}$ is not contained in Kropholler’s class ${\scriptstyle{\rm H}}\mathfrak F$.