Volumes of hyperbolic three-manifolds associated to modular links

Alex Brandts; Tali Pinsky; Lior Silberman

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arXiv · 1705.04760

Volumes of hyperbolic three-manifolds associated to modular links

Alex Brandts·Tali Pinsky·Lior Silberman

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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

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Alex Brandts, Tali Pinsky, Lior Silberman. 2017-05-12. Volumes of hyperbolic three-manifolds associated to modular links. https://arxiv.org/abs/1705.04760

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