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 $\pi ⋊G$ 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.