$\mathbb{Z}_p$-torus actions on positively curved manifolds
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908816592863232 |
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| author | Abdullah, Muhammad Searle, Catherine |
| author_facet | Abdullah, Muhammad Searle, Catherine |
| contents | In this article, we study closed, positively curved $n$-manifolds that admit an effective, isometric $\mathbb{Z}_p^r$-action with a fixed point, where $p$ is an odd prime. For all sufficiently large $n$, we obtain a symmetry-rank bound in Theorem A that improves the $3n/8$ bound of Fang and Rong and of Ghazawneh. We improve on this bound for small odd primes $3\leq p\leq 19$ in Theorem B. One of our main tools comes from the theory of error-correcting codes and is of independent interest: we derive a finite-length Plotkin bound and a finite-length Elias-Bassalygo bound for $q$-ary codes and show that the finite-length Plotkin bound is asymptotically sharper for all primes $p \ge 23$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_26853 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | $\mathbb{Z}_p$-torus actions on positively curved manifolds Abdullah, Muhammad Searle, Catherine Differential Geometry 53C21, 20K01 In this article, we study closed, positively curved $n$-manifolds that admit an effective, isometric $\mathbb{Z}_p^r$-action with a fixed point, where $p$ is an odd prime. For all sufficiently large $n$, we obtain a symmetry-rank bound in Theorem A that improves the $3n/8$ bound of Fang and Rong and of Ghazawneh. We improve on this bound for small odd primes $3\leq p\leq 19$ in Theorem B. One of our main tools comes from the theory of error-correcting codes and is of independent interest: we derive a finite-length Plotkin bound and a finite-length Elias-Bassalygo bound for $q$-ary codes and show that the finite-length Plotkin bound is asymptotically sharper for all primes $p \ge 23$. |
| title | $\mathbb{Z}_p$-torus actions on positively curved manifolds |
| topic | Differential Geometry 53C21, 20K01 |
| url | https://arxiv.org/abs/2510.26853 |