Volumes of Hyperbolic Three-Manifolds Associated with Modular Links

Volumes of Hyperbolic Three-Manifolds Associated with Modular Links

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 al...

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Veröffentlicht in:Symmetry (Basel) 2019-10, Vol.11 (10), p.1206
Hauptverfasser: Brandts, Alex, Pinsky, Tali, Silberman, Lior
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Fields (mathematics)
Geodesy
Manifolds
Number theory
Orbits
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Pinsky, Tali
Silberman, Lior
description 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.
doi_str_mv 10.3390/sym11101206
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subjects Fields (mathematics)
Geodesy
Manifolds
Number theory
Orbits
title Volumes of Hyperbolic Three-Manifolds Associated with Modular Links
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