Equivariant dimensions of groups with operators

Abstract

Let $\pi$ be a group equipped with an action of a second group $G$ by automorphisms. We define the equivariant cohomological dimension $\mathsf{cd}\_G(\pi)$, the equivariant geometric dimension $\mathsf{cat}\_G(\pi)$, and the equivariant Lusternik–Schnirelmann category $\mathsf{gd}\_G(\pi)$ in terms of the Bredon dimensions and classifying space of the family of subgroups of the semi-direct product $\pi\rtimes G$ consisting of sub-conjugates of $G$. When $G$ is finite, we extend theorems of Eilenberg–Ganea and Stallings–Swan to the equivariant setting, thereby showing that all three invariants coincide (except for the possibility of a $G$-group $\pi$ with $\mathsf{cat}\_G(\pi)=\mathsf{cd}\_G(\pi)=2$ and $\mathsf{gd}\_G(\pi)=3$). A main ingredient is the purely algebraic result that the cohomological dimension of any finite group with respect to any family of proper subgroups is greater than one. This implies a Stallings–Swan type result for families of subgroups which do not contain all finite subgroups.