Universal virtual braid groups

We introduce the universal virtual braid group $U{V}_n(c)$, 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 $U{V}_n(c)$ 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 $U{V}_n(c{)}^{\mathrm{\prime }}$ 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 $U{V}_n(c)$ and its pure subgroup $PU{V}_n(c)$, 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 $U{V}_n(c)$ whose order is strictly larger than $n!$.