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