Universal virtual braid groups

TL;DR

This paper introduces the universal virtual braid group $UV_n(c)$, unifying various virtual braid structures, and explores its algebraic properties, subgroup structure, and quotients.

Contribution

It defines the universal virtual braid group, proves its key structural properties, and classifies subgroup separability and the Howson property.

Findings

$UV_n(c)$ contains a finite-index right-angled Artin subgroup.

Residual finiteness, linearity, and solvability of word and conjugacy problems are established.

Classified subgroup separability and the Howson property for $UV_n(c)$ and $PUV_n(c)$.

Abstract

We introduce the universal virtual braid group $UV_n(c)$, which provides a unified algebraic framework for virtual braid--type structures with $c$ types of crossings and admits natural quotient maps onto the standard families in the literature. We prove that $UV_n(c)$ contains a right-angled Artin subgroup of finite index, yielding strong structural consequences: residual finiteness, linearity, solvability of the word and conjugacy problems, and the Tits alternative. For $n\ge 5$, the commutator subgroup $UV_n(c)'$ is perfect, and every non-abelian finite image contains a subgroup isomorphic to the symmetric group $S_n$; in particular, $S_n$ is the smallest non-abelian finite quotient. These rigidity phenomena persist under a broad class of natural quotients, including virtual braid, virtual singular braid, virtual twin and multi-virtual braid groups. We further obtain a complete classification of subgroup separability (LERF) and the Howson property for $UV_n(c)$ and its pure subgroup $PUV_n(c)$, showing that both properties hold precisely for $n\le 3$. We also compute the virtual cohomological dimension, determine the center, prove that the finite-index RAAG subgroup is characteristic, and construct explicit finite quotients of $UV_n(c)$ whose order is strictly larger than $n!$.