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