| Summary: | Periodic geodesics on the modular surface correspond to periodic orbits of the geodesic flow in its unit tangent bundle <inline-formula> <math display="inline"> <semantics> <mrow> <msub> <mi>PSL</mi> <mn>2</mn> </msub> <mrow> <mrow> <mo>(</mo> <mi mathvariant="double-struck">Z</mi> <mo>)</mo> </mrow> <mo>∖</mo> </mrow> <msub> <mi>PSL</mi> <mn>2</mn> </msub> <mrow> <mo>(</mo> <mi mathvariant="double-struck">R</mi> <mo>)</mo> </mrow> </mrow> </semantics> </math> </inline-formula>. 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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