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

Abstract

In this paper we show that most rank two groups act freely on a finite homotopy product of two spheres. This makes new progress on a conjecture by Benson and Carlson which states 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. Recalling that Qd(p) is the semidirect product of a rank two elementary abelian p-group with SL(2,p), we show that a rank two finite group acts freely on a finite CW-complex homotopic to the product of two spheres if, it does not contain any subquotient isomorphic to Qd(p) for any odd prime p.