Systoles of hyperbolic manifolds

Abstract

We show that for every $n\ge 2$ and any $ϵ>0$ there exists a compact hyperbolic $n$–manifold with a closed geodesic of length less than $ϵ$. When $ϵ$ is sufficiently small these manifolds are non-arithmetic, and they are obtained by a generalised inbreeding construction which was first suggested by Agol for $n=4$. We also show that for $n\ge 3$ the volumes of these manifolds grow at least as $1∕{ϵ}^{n-2}$ when $ϵ\to 0$.