Congruence subgroups from representations of the three-strand braid group

TL;DR

This paper investigates when kernels of certain braid group representations form congruence subgroups, revealing dimension-dependent criteria and constructing explicit non-congruence examples in three dimensions.

Contribution

It establishes conditions under which braid group representations yield congruence subgroups and constructs explicit non-congruence examples in three dimensions.

Findings

In dimensions two and three, certain projective orders imply congruence subgroups.

For three-dimensional representations, projective order alone does not determine the congruence property.

Explicit non-congruence subgroups are constructed for specific projective orders in three dimensions.

Abstract

Ng and Schauenburg proved that the kernel of a $(2+1)$-dimensional topological quantum field theory representation of $\mathrm{SL}(2, \mathbb{Z})$ is a congruence subgroup. Motivated by their result, we explore when the kernel of an irreducible representation of the braid group $B_3$ with finite image enjoys a congruence subgroup property. In particular, we show that in dimensions two and three, when the projective order of the image of the braid generator $\sigma_1$ is between 2 and 5 the kernel projects onto a congruence subgroup of $\mathrm{PSL}(2,\mathbb{Z})$ and compute its level. However, we prove for three dimensional representations, the projective order is not enough to decide the congruence property. For each integer of the form $2\ell \geq 6$ with $\ell$ odd, we construct a pair of non-congruence subgroups associated with three-dimensional representations having finite image and $\sigma_1$ mapping to a matrix with projective order $2\ell$. Our technique uses classification results of low dimensional braid group representations, and the Fricke-Wohlfarht theorem in number theory.