A G-space is an E_fin(G)-space if it a G-CW-complex whose fixed sets E_fin(G)^H are contractible when H is a finite subgroup of G and are empty otherwise. It is not difficult to show that an two E_fin(G)-spaces are G-homotopy equivalent. The question of equivariant rigidity asks if, for a given group G, whether any two cocompact E_fin(G)-manifolds are G-homeomorphic. For a torsion-free group this is equivalent to the Borel Conjecture for G. In joint work with Connolly and Khan, we give a complete analysis of equivariant rigidity in the case where the normalizer of each nontrivial finite subgroup is finite. We are currently studying the case where the normalizers have higher dimensional vcd’s.

In joint work with Farrell, we are studying the corresponding uniqueness question - if there is a finite index torsion free normal subgroup G_0 whose K(G,1) has the homotopy type of a closed manifold, is there a cocompact manifold model for E_{fin}G? This is closely related to the Nielsen realization question.