Free Actions of Finite Groups on Sⁿ × Sⁿ

Original · EN

Let p be an odd prime. We construct a non-abelian extension Γ of S¹ 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.