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

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.

arXiv commentOur preprint "Free actions of extraspecial p-groups on S^n x S^n" (arXiv:math/0701558) is now divided into two separate papers. This is the final version of the second part - to appear in Transactions AMS