Free Actions of Finite Groups on $S^n \times S^n$

TL;DR

This paper constructs specific non-abelian groups and demonstrates their free, smooth actions on products of spheres, advancing understanding of finite group actions on high-dimensional manifolds.

Contribution

It introduces a new method to establish free actions of certain non-metacyclic p-groups on products of spheres, expanding known classes of group actions.

Findings

Existence of free smooth actions of non-metacyclic p-groups on $S^{2p-1} imes S^{2p-1}$

Construction of a non-abelian extension of $S^1$ by $Z/p imes Z/p$

Application of a general approach to group actions on products of spheres

Abstract

Let $p$ be an odd prime. We construct a non-abelian extension $\Gamma$ of $S^1$ by $Z/p \times Z/p$, and prove that any finite subgroup of $\Gamma$ acts freely and smoothly on $S^{2p-1} \times S^{2p-1}$. In particular, for each odd prime $p$ we obtain free smooth actions of infinitely many non-metacyclic rank two $p$-groups on $S^{2p-1} \times S^{2p-1}$. These results arise from a general approach to the existence problem for finite group actions on products of equidimensional spheres.