This work uncovers the pure virtual braid group as a level 2 congruence subgroup in virtual braid groups, suggesting parallels to classical structures.
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.
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Bloomquist et al. (2026) studied this question.
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