Volume 11, issue 3 (2011)

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ISSN (electronic): 1472-2739
ISSN (print): 1472-2747
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 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 1ϵn2 when ϵ 0.

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