Qd(p)-free rank two finite groups act freely on a homotopy product of two spheres

A classic result of Swan states that a finite group GG acts freely on a finite homotopy sphere if and only if every abelian subgroup of GG is cyclic. Following this result, Benson and Carlson conjectured that a finite group GG acts freely on a finite complex with the homotopy type of nn spheres if the rank of GG is less than or equal to nn. Recently, Adem and Smith have shown that every rank two finite pp-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 Qd(p)-free act freely on a finite homotopy product of two spheres.