Equivariant dimensions of groups with operators

Abstract

Let $\ddot{I}€$ be a group equipped with an action of a second group $G$ by automorphisms. We define the equivariant cohomological dimension ${\mathsf{cd}}_G(\ddot{I}€)$, the equivariant geometric dimension ${\mathsf{gd}}_G(\ddot{I}€)$, and the equivariant Lusternik-Schnirelmann category ${\mathsf{cat}}_G(\ddot{I}€)$ in terms of the Bredon dimensions and classifying space of the family of subgroups of the semi-direct product $\ddot{I}€⋊G$ 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 $\ddot{I}€$ with ${\mathsf{cat}}_G(\ddot{I}€)={\mathsf{cd}}_G(\ddot{I}€)=2$ and ${\mathsf{gd}}_G(\ddot{I}€)=3$). 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.

v3: 23 pages. Added Remark 2.7 and strengthened Example 4.3