Equivariant dimensions of groups with operators

Let $\pi$ be a group equipped with an action of second $G$ by automorphisms. We define the equivariant cohomological dimension $\mathsf{cd}\_G(\pi)$, geometric $\mathsf{cat}\_G(\pi)$, and Lusternik–Schnirelmann category $\mathsf{gd}\_G(\pi)$ in terms Bredon dimensions classifying space family subgroups semi-direct product $\pi\rtimes G$ consisting sub-conjugates $G$. When is finite, we extend theorems Eilenberg–Ganea Stallings–Swan to setting, thereby showing that all three invariants coincide (except for possibility $G$-group $\mathsf{cat}\_G(\pi)=\mathsf{cd}\_G(\pi)=2$ $\mathsf{gd}\_G(\pi)=3$). A main ingredient purely algebraic result any finite respect proper greater than one. This implies type families which do not contain subgroups.