Density of systoles of hyperbolic manifolds

Abstract

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

Minor edits, as per the referee's comments. Final version, to appear in Comptes Rendus Mathématique