# Volumes of hyperbolic three-manifolds associated to modular links

## TL;DR

This paper investigates the hyperbolic volumes of 3-manifolds associated with modular links, providing numerical evidence that volumes grow linearly with geodesic length for certain algebraic sets, but not universally.

## Contribution

It introduces a novel numerical analysis of hyperbolic 3-manifolds linked to modular geodesics, highlighting specific asymptotic behaviors related to algebraic structures.

## Key findings

- Volume exhibits linear growth with geodesic length for ideal class group geodesics
- Different sets of geodesics do not necessarily show linear volume growth
- Numerical evidence supports conjectures about geometric properties of 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