Systoles of hyperbolic manifolds

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

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Keywords

systole, hyperbolic manifold, nonarithmetic lattice

Mathematical Subject Classification 2010

Primary: 22E40, 53C22

References

Publication

Received: 4 October 2010

Revised: 24 January 2011

Accepted: 12 February 2011

Published: 17 May 2011

Authors

Mikhail V Belolipetsky
Department of Mathematical Sciences Durham University South Road Durham DH1 3LE United Kingdom	Institute of Mathematics Koptyuga 4 630090 Novosibirsk Russia

Scott A Thomson
Department of Mathematical Sciences Durham University South Road Durham DH1 3LE United Kingdom

Volume 11, issue 3 (2011)

Mikhail V Belolipetsky and Scott A Thomson

Abstract