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Universal virtual braid groups

Oscar Ocampo

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 $n!$.

Universal virtual braid groups

Abstract

We introduce the universal virtual braid group , which provides a unified algebraic framework for virtual braid--type structures with types of crossings and admits natural quotient maps onto the standard families in the literature. We prove that 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 , the commutator subgroup is perfect, and every non-abelian finite image contains a subgroup isomorphic to the symmetric group ; in particular, 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 and its pure subgroup , showing that both properties hold precisely for . 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 whose order is strictly larger than .

Paper Structure

This paper contains 18 sections, 37 theorems, 44 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The universal virtual braid group
  3. Definition
  4. Natural quotients and maps to $S_n$
  5. Inheritance of the semidirect structure
  6. A finite index right--angled Artin subgroup
  7. Structural and residual properties
  8. Properties of universal virtual braid groups
  9. Virtual cohomological dimension
  10. Center of $UV_n(c)$
  11. Subgroup separability
  12. The Howson property
  13. Finite quotients and rigidity phenomena
  14. Perfectness of the commutator and stability under quotients
  15. Non--abelian finite quotients
  16. ...and 3 more sections

Key Result

Theorem 1

For every $n\ge 2$ and $c\ge 1$, the group $UV_n(c)$ contains a right--angled Artin subgroup of finite index. $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure 1: A braid with 4 strands and $c=2$ types of (non-virtual) crossings. Virtual crossings are encircled by a small circle.

Theorems & Definitions (81)

  • Theorem 1
  • Theorem 2
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • proof
  • Proposition 2.6
  • ...and 71 more