Equivariant dimensions of groups with operators

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