#### Volume 11, issue 3 (2011)

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Systoles of hyperbolic manifolds

### Mikhail V Belolipetsky and Scott A Thomson

Algebraic & Geometric Topology 11 (2011) 1455–1469
##### 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 $ϵ\to 0$.

##### Keywords
systole, hyperbolic manifold, nonarithmetic lattice
##### Mathematical Subject Classification 2010
Primary: 22E40, 53C22
##### 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