Density of systoles of hyperbolic manifolds

Sami Douba; Junzhi Huang

doi:10.5802/crmath.689

Article de recherche - Géométrie et Topologie, Théorie des groupes

Density of systoles of hyperbolic manifolds
[Densité de systoles de variétés hyperboliques]

Sami Douba ¹ ; Junzhi Huang ²

¹ Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France
² Department of Mathematics, Yale University, New Haven, CT 06511, USA

Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1819-1824.

Résumé
Abstract

Nous démontrons que, pour tout $n \geq 2$ , les systoles de variétés hyperboliques compactes sans bord de dimension $n$ constituent une partie dense de $] 0, + \infty [$ . Nous démontrons de plus que, pour tout $n \geq 2$ et tout nombre de Salem $λ$ , il existe une variété hyperbolique arithmétique compacte sans bord de dimension $n$ et de systole $log (λ)$ . En particulier, la conjecture de Salem est vraie si et seulement si les systoles de variétés hyperboliques arithmétiques compactes sans bord d’une certaine dimension (de manière équivalente, de dimension quelconque) ne sont pas denses dans $] 0, + \infty [$ .

We show that for each $n \geq 2$ , the systoles of closed hyperbolic $n$ -manifolds form a dense subset of $(0, + \infty)$ . We also show that for any $n \geq 2$ and any Salem number $λ$ , there is a closed arithmetic hyperbolic $n$ -manifold of systole $log (λ)$ . 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 $(0, + \infty)$ .

Reçu le : 2024-06-05
Accepté le : 2024-09-17
Publié le : 2024-11-25

DOI : 10.5802/crmath.689

Classification : 22E40, 11F06
Keywords: Geometric topology, hyperbolic manifolds, systoles, arithmetic groups
Mots-clés : Topologie géométrique, variétés hyperboliques, systoles, groupes arithmétiques

Affiliations des auteurs :

Sami Douba ¹ ; Junzhi Huang ²

¹ Institut des Hautes Études Scientifiques, 35 route de Chartres, 91440 Bures-sur-Yvette, France
² Department of Mathematics, Yale University, New Haven, CT 06511, USA

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     author = {Sami Douba and Junzhi Huang},
     title = {Density of systoles of hyperbolic manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {1819--1824},
     publisher = {Acad\'emie des sciences, Paris},
     volume = {362},
     year = {2024},
     doi = {10.5802/crmath.689},
     language = {en},
}

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Sami Douba; Junzhi Huang. Density of systoles of hyperbolic manifolds. Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1819-1824. doi : 10.5802/crmath.689. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.689/

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