Congruence Subgroups of the Virtual Braid Group
Wade Bloomquist, Alexa Goldberg, Nancy Scherich
TL;DR
This work extends the notion of congruence subgroups from the classical braid group to the virtual braid group via an extension of the integral Burau representation, introducing $vB_n[m]=\ker(r_m\circ\rho_v)$ and comparing it with $B_n[m]$. The authors provide an algebraic proof of Arnol\'d\’s classical result that $B_n[2]=P_n$, and prove the virtual analogue $vB_n[2]=vP_n$, along with structural properties such as $B_n[m]\subsetneq vB_n[m]\subsetneq vB_n$, $vB_n[m]\cap vB_n[\ell]=vB_n[\operatorname{lcm}(m,\ell)]$, and the relation of images to dihedral groups in low rank, especially $vB_2$. They also analyze the case $n=2$ in depth, deriving a short exact sequence $1\to vB_2[m]\to vB_2\to D_{2m}\to1$ and explicit normal generators for $vB_2[m]$, and discuss questions about when $vB_n/vB_n[m]$ is a subgroup of $vB_n$, as well as potential extensions to welded braids. The results reveal key differences between classical and virtual congruence subgroups and point to several open problems in higher rank.
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.