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Volumes of Hyperbolic Three-Manifolds Associated with Modular Links Brandts, Alex; Pinsky, Tali; Silberman, Lior

Abstract

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL₂(ℤ)\PSL₂(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.

Item Metadata

Title
Volumes of Hyperbolic Three-Manifolds Associated with Modular Links
Creator
Brandts, Alex; Pinsky, Tali; Silberman, Lior
Publisher
Multidisciplinary Digital Publishing Institute
Date Issued
2019-09-26
Description
Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle PSL₂(ℤ)\PSL₂(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.
Subject
modular group; primitive geodesics; hyperbolic volume
Genre
Article
Type
Text
Language
eng
Date Available
2019-10-25
Provider
Vancouver : University of British Columbia Library
Rights
CC BY 4.0
DOI
10.14288/1.0384851
URI
http://hdl.handle.net/2429/72059
Affiliation
Science, Faculty of; Non UBC; Mathematics, Department of
Citation
Symmetry 11 (10): 1206 (2019)
Publisher DOI
10.3390/sym11101206
Peer Review Status
Reviewed
Scholarly Level
Faculty
Rights URI
https://creativecommons.org/licenses/by/4.0/
Aggregated Source Repository
DSpace

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CC BY 4.0