Examples of Free Actions on Products of Spheres

We construct a non-abelian extension $\Gamma$ of $S^1$ by $\cy 3 \times \cy 3$, and prove that $\Gamma$ acts freely and smoothly on $S^{5} \times S^{5}$. This gives new actions on $S^{5} \times S^{5}$ for an infinite family $\cP$ of finite 3-groups. We also show that any finite odd order subgroup of the exceptional Lie group $G_2$ admits a free smooth action on $S^{11}\times S^{11}$. This gives new actions on $S^{11}\times S^{11}$ for an infinite family $\cE $ of finite groups. We explain the significance of these families $\cP $, $\cE $ for the general existence problem, and correct some mistakes in the literature.

💡 Research Summary

The paper addresses the long‑standing problem of constructing free smooth actions of finite groups on products of spheres, focusing on the previously under‑explored cases where the sphere dimensions are odd. The authors present two major families of groups that admit such actions, together with explicit constructions and a correction of errors in earlier literature.

First, they build a non‑abelian central extension
\