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Open AccessArticle

Volumes of Hyperbolic Three-Manifolds Associated with Modular Links

Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK
Department of Mathematics, The Technion, Haifa 3200003, Israel
Department of Mathematics, The University of British Columbia, Vancouver, BC V6T 1Z4, Canada
Author to whom correspondence should be addressed.
These authors contributed equally to this work.
Symmetry 2019, 11(10), 1206;
Received: 14 August 2019 / Revised: 18 September 2019 / Accepted: 19 September 2019 / Published: 26 September 2019
Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL 2 ( Z ) PSL 2 ( R ) . A finite collection of such orbits is a collection of disjoint closed curves in a 3-manifold, in other words a link. The complement of those links is always a hyperbolic 3-manifold, and hence has a well-defined volume. We present strong numerical evidence that, in the case of the set of geodesics corresponding to the ideal class group of a real quadratic field, the volume has linear asymptotics in terms of the total length of the geodesics. This is not the case for general sets of geodesics. View Full-Text
Keywords: modular group; primitive geodesics; hyperbolic volume modular group; primitive geodesics; hyperbolic volume
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Brandts, A.; Pinsky, T.; Silberman, L. Volumes of Hyperbolic Three-Manifolds Associated with Modular Links. Symmetry 2019, 11, 1206.

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