Density of systoles of hyperbolic manifolds

TL;DR

This paper proves that the set of systoles of closed hyperbolic n-manifolds is dense in positive real numbers and connects this to Salem numbers and the Salem conjecture, revealing deep geometric and number-theoretic links.

Contribution

It establishes the density of systoles in hyperbolic manifolds and links systole lengths to Salem numbers, providing new insights into their distribution and arithmetic properties.

Findings

Systoles of closed hyperbolic n-manifolds are dense in (0, +∞).

Existence of hyperbolic n-manifolds with systole log(λ) for any Salem number λ.

The Salem conjecture relates to the density of systoles in arithmetic hyperbolic manifolds.

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 $(0, +\infty)$.