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

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.

💡 Research Summary

The paper addresses the long‑standing problem of determining which finite groups can act freely and smoothly on a product of two equidimensional spheres, (S^{n}\times S^{n}). While many results are known for cyclic, metacyclic, or orthogonal groups, the case of non‑metacyclic rank‑two (p)-groups has remained largely unexplored. The authors introduce a new construction that produces an infinite family of such groups and proves that every finite subgroup of the constructed ambient group acts freely on (S^{2p-1}\times S^{2p-1}) for any odd prime (p).

The central object is a non‑abelian central extension (\Gamma) of the circle group (S^{1}) by (\mathbb Z/p\times\mathbb Z/p). Explicitly, (\Gamma) fits into the short exact sequence
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