Free actions of finite groups on $S^n \times S^n$

Abstract

Let p be an odd prime. We construct a non-abelian extension Γ of S by Z/p × Z/p, and prove that any finite subgroup of Γ acts freely and smoothly on S2p−1 × S2p−1. In particular, for each odd prime p we obtain free smooth actions of infinitely many non-metacyclic rank two p-groups on S2p−1 × S2p−1. These results arise from a general approach to the existence problem for finite group actions on products of equidimensional spheres.