Free actions of finite groups on 𝑆ⁿ×𝑆ⁿ

Hambleton, Ian; Ünlü, Özgün

doi:10.1090/S0002-9947-09-05039-9

Free actions of finite groups on $S^n \times S^n$
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by Ian Hambleton and Özgün Ünlü PDF

Trans. Amer. Math. Soc. 362 (2010), 3289-3317 Request permission

Abstract:

Let $p$ be an odd prime. We construct a non-abelian extension $\Gamma$ of $S^1$ by $\mathbf {Z}/p \times \mathbf {Z}/p$, 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.

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