Systoles of hyperbolic manifolds

Abstract

We show that for every n ≥ 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 ≥ 3 the volumes of these manifolds grow at least as 1/ ε n-2 when ε->0.