image-free rank two finite groups act freely on a homotopy product of two spheres

A classic result of Swan states that a finite group G acts freely on a finite homotopy sphere if and only if every abelian subgroup of G is cyclic. Following this result, Benson and Carlson conjectured that a finite group G acts freely on a finite complex with the homotopy type of n spheres if the rank of G is less than or equal to n. Recently, Adem and Smith have shown that every rank two finite p-group acts freely on a finite complex with the homotopy type of two spheres. In this paper we will make further progress, showing that rank two groups that are image-free act freely on a finite homotopy product of two spheres.