Density of systoles of hyperbolic manifolds

Douba, Sami; Huang, Junzhi

Comptes Rendus. Mathématique, Tome 362 (2024) no. G12, pp. 1819-1824

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 $log\left(\lambda \right)$. 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 $(0,+\infty )$.

Nous démontrons que, pour tout $n\ge 2$, 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 $n\ge 2$ et tout nombre de Salem $\lambda$, il existe une variété hyperbolique arithmétique compacte sans bord de dimension $n$ et de systole $log\left(\lambda \right)$. 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 $]0,+\infty [$.

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

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