Statistics of finite degree covers of torus knot complements

TL;DR

This paper investigates the asymptotic subgroup growth and properties of random finite covers of torus knot complements, revealing their limiting behaviors and extending results to broader classes of lattices in hyperbolic geometry.

Contribution

It provides new asymptotic results on subgroup growth and random covers of torus knot complements, including their Benjamini-Schramm limits and Betti number growth, generalizing to larger lattice classes.

Findings

Determined the asymptotic subgroup growth of torus knot complement groups.

Established the Benjamini-Schramm limit for random finite covers.

Proved linear growth of Betti numbers in these covers.

Abstract

In the first part of this paper, we determine the asymptotic subgroup growth of the fundamental group of a torus knot complement. In the second part, we use this to study random finite degree covers of torus knot complements. We determine their Benjamini-Schramm limit and the linear growth rate of the Betti numbers of these covers. All these results generalise to a larger class of lattices in $\mathrm{PSL}(2,\mathbb{R})\times \mathbb{R}$. As a by-product of our proofs, we obtain analogous limit theorems for high degree random covers of non-uniform Fuchsian lattices with torsion.