Universal virtual braid groups
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866911577927581696 |
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| author | Ocampo, Oscar |
| author_facet | Ocampo, Oscar |
| contents | 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 $n!$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_01633 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Universal virtual braid groups Ocampo, Oscar Group Theory Geometric Topology 20F36, 20F65, 20E26 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 $n!$. |
| title | Universal virtual braid groups |
| topic | Group Theory Geometric Topology 20F36, 20F65, 20E26 |
| url | https://arxiv.org/abs/2604.01633 |