Free Actions of Extraspecial $p$-Groups on $S^n \times S^n$

Abstract

Let $p$ be an odd regular prime, and let $G_p$ denote the extraspecial $p$--group of order $p^{3}$ and exponent $p$. We show that $G_p$ acts freely and smoothly on $S^{2p-1} \times S^{2p-1}$. For $p=3$ we explicitly construct a free smooth action of a Lie group $\widetilde{G}_3$ containing $G_3$ on $S^{5} \times S^{5}$. In addition, we show that any finite odd order subgroup of the exceptional Lie group $\Gtwo $ admits a free smooth action on $S^{11}\times S^{11}$.