Most 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 act freely on a finite homotopy product of two spheres. By letting Tp be the semidirect product (Zp ×Zp)⋊θ SL2(Fp) where the action θ is given by the obvious inclusion SL2(Fp) → GL2(Fp) ∼= Aut(Zp×Zp), we will show that a rank two finite group acts freely on a finite homotopy product of two spheres if, for each prime p, it does not contain a subgroup H such that H/Op′(H) ∼= Tp. 2000 MSC: 57Q91, 55R25, 20C15, 20E15