Equivariant Trees and Partition Complexes

We introduce two definitions of \(G\)-equivariant partitions of a finite \(G\)-set, both of which yield \(G\)-equivariant partition complexes. By considering suitable notions of equivariant trees, we show that \(G\)-equivariant partitions and \(G\)-trees are \(G\)-homotopy equivalent, generalizing existing results for the non-equivariant setting. Along the way, we develop equivariant versions of Quillen's Theorems A and B, which are of independent interest.