Density of systoles of hyperbolic manifolds
We show that for each $n \ge 2$, the systoles of closed hyperbolic $n$-manifolds form a dense subset of $(0, +\infty )$. We also show that for any $n\ge 2$ and any Salem number $\lambda $, there is a closed arithmetic hyperbolic $n$-manifold of systole $\log (\lambda )$. In particular, the Salem con...
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| Main Authors: | , |
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| Format: | Article |
| Language: | English |
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Académie des sciences
2024-11-01
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| Series: | Comptes Rendus. Mathématique |
| Subjects: | |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.689/ |
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| Summary: | We show that for each $n \ge 2$, the systoles of closed hyperbolic $n$-manifolds form a dense subset of $(0, +\infty )$. We also show that for any $n\ge 2$ and any Salem number $\lambda $, there is a closed arithmetic hyperbolic $n$-manifold of systole $\log (\lambda )$. In particular, the Salem conjecture holds if and only if the systoles of closed arithmetic hyperbolic manifolds in some (any) dimension fail to be dense in $(0, +\infty )$. |
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| ISSN: | 1778-3569 |