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Open AccessArticle

Volumes of Hyperbolic Three-Manifolds Associated with Modular Links

1
Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK
2
Department of Mathematics, The Technion, Haifa 3200003, Israel
3
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; https://doi.org/10.3390/sym11101206
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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