  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
