Equivariant dimensions of groups with operators

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

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