Congruence Subgroups of the Virtual Braid Group

TL;DR

This paper extends the concept of congruence subgroups to the virtual braid group via an extended Burau representation, establishing a key subgroup equivalence and exploring differences from classical braid groups.

Contribution

It introduces a new framework for congruence subgroups in the virtual braid group and proves the pure virtual braid group is the level 2 congruence subgroup.

Findings

Level 2 congruence subgroup is the pure virtual braid group

Established a virtual analogue of Arnol'd's classical result

Highlighted differences between classical and virtual braid groups

Abstract

We extend the notion of congruence subgroups of the braid group to the virtual braid group using an extension of the integral Burau representation. We prove that the level 2 congruence subgroup of the virtual braid group is the pure virtual braid group, recovering a virtual analogue of a result of Arnol'd. We pose several questions which highlight the difference between the classical and virtual braid groups.