Comptes Rendus
Research article - Geometry and Topology, Group theory
Density of systoles of hyperbolic manifolds
Comptes Rendus. Mathématique, Volume 362 (2024), pp. 1819-1824.

We show that for each n2, the systoles of closed hyperbolic n-manifolds form a dense subset of (0,+). We also show that for any n2 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,+).

Nous démontrons que, pour tout n2, les systoles de variétés hyperboliques compactes sans bord de dimension n constituent une partie dense de ]0,+[. Nous démontrons de plus que, pour tout n2 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,+[.

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Published online:
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

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
License: CC-BY 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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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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