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Let $\pi$ be a group equipped with an action of a second group $G$ byautomorphisms. We define the equivariant cohomological dimension ${\sfcd}_G(\pi)$, the equivariant geometric dimension ${\sf gd}_G(\pi)$, and theequivariant Lusternik-Schnirelmann category ${\sf cat}_G(\pi)$ in terms of theBredon dimensions and classifying space of the family of subgroups of thesemi-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 theequivariant setting, thereby showing that all three invariants coincide (exceptfor the possibility of a $G$-group $\pi$ with ${\sf cat}_G(\pi)={\sfcd}_G(\pi)=2$ and ${\sf gd}_G(\pi)=3$). A main ingredient is the purelyalgebraic result that the cohomological dimension of any finite group withrespect to any family of proper subgroups is greater than one. This implies aStallings-Swan type result for families of subgroups which do not contain allfinite subgroups.

Equivariant dimensions of groups with operators