Systoles of hyperbolic manifolds

By Mikhail V. Belolipetsky, Scott Thomson. 2010

54 citations recorded by the source corpus.

We show that for every $n\geq 2$ 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 $n\geq 3$ the volumes of these manifolds grow at least as $1/\epsilon^{n-2}$ when $\epsilon\to 0$.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/2e239d7225b0884b7e6cf8240958a877361f22a3

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In Geometry and Topology

In Mathematical Physics

Citations in the corpus

On the growth of $L^2$-invariants for sequences of lattices in Lie groups (2012, background): “For X = Hd, it is known that for all d, there are closed hyperbolic d-manifolds with arbitrarily short systoles, [9], [22], [16].”