Systoles of Hyperbolic Manifolds

Abstract

We show that for every $n\ge 2$ and any $\epsilon >0$ there exists a compact hyperbolic $n$-manifold with a closed geodesic of length less than $\epsilon$. When $\epsilon$ 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 $1/{\epsilon }^{n-2}$ when $\epsilon \to 0$.

12 pages; revised following referee's comments; to appear in Algebraic and Geometric Topology

References in corpus (1)

• Noncoherence of some lattices in Isom(Hn)

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