Volumes of Hyperbolic Three-Manifolds Associated with Modular Links

Abstract

Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle $\\mathrm{PSL}_2(\\mathbb{Z})\\backslash\\mathrm{PSL}_2(\\mathbb{R})$. The complement of any finite number of orbits is a hyperbolic $3$-manifold, which thus 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