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Mathematics > Geometric Topology
Title: Density of systoles of hyperbolic manifolds
(Submitted on 24 Apr 2024)
Abstract: 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 $\lambda$, 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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