Density of systoles of hyperbolic manifolds

2024/04/24 by Sami Douba, Douba, Sami, Junzhi Huang +1
Computer Science · Mathematics · #Computational Geometry and Mesh Generation #FOS: Mathematics #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Group Theory (math.GR) #Mathematical Dynamics and Fractals #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2404.15927

openalex publication_date 2024/04/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that for each n ≥ 2, the systoles of closed hyperbolic n-manifolds form a dense subset of (0, +∞). We also show that for any n≥ 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, +∞).

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