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 cdG(π), the equivariant geometric dimension gdG(π), and the equivariant Lusternik-Schnirelmann category catG(π) in terms of the Bredon dimensions and classifying space of the family of subgroups of the semi-direct product π⋊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 catG(π)=cdG(π)=2 and gdG(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.