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Oscar Ocampo

Publications and source records attributed to Oscar Ocampo.

At least 19 recordsLinked to original sources
arXiv2026

Forbidden relations in universal virtual braid groups

We study natural automorphisms of the universal virtual braid group $UV_n(k)$. These automorphisms induce commuting involutions in the outer automorphism group and generate a subgroup isomorphic to $\mathbb{Z}_2^k\times\mathbb{Z}_2$. We then show that the two one-forbidden quotients of $UV_n(k)$ are isomorphic. Furthermore, we introduce the universal unrestricted virtual braid group $UUV_n(k)$ obtained by imposing simultaneously the two forbidden relations, and derive several structural properties inherited from the universal setting. Since the multi-virtual braid group $M_kVB_n$ is a quotient of $UV_n(k)$, the corresponding results for $M_kVB_n$ follow as consequences. In particular, for $k=1$ we prove that the quotients of $VB_n$ by the two forbidden relations are isomorphic and obtain structural properties for the unrestricted virtual braid group.

math.GR↗
arXiv2026

Multi-welded twin groups

For $k\geq 1$ and $n\geq 2$, we introduce the multi-welded twin group $M_kWT_n$, a natural welded analogue of the multi-virtual twin group. We show that $M_kWT_n$ arises naturally as a quotient of the universal welded braid group $UW_n(k)$, placing it within the unified framework of universal virtual and welded braid-type groups. We establish natural quotient maps relating $M_kWT_n$ to the multi-virtual twin group $M_kVT_n$, the welded twin group $WT_n$, and the corresponding virtual and welded braid-type groups. Several structural properties of $M_kWT_n$ are obtained. In particular, we compute its abelianization, prove that its commutator subgroup is perfect for $n\ge5$, and show that the symmetric group $S_n$ is its smallest non-abelian finite quotient. We also investigate the representation theory of $M_kWT_n$. In fact, we classify all non-trivial complex homogeneous $2$-local representations of $M_kWT_n$, showing that only one family survives under the additional twin and welded relations. Furthermore, we classify all non-trivial complex homogeneous $3$-local representations of $M_2WT_n$. We further investigate the reducibility and faithfulness properties of both the $2$-local and $3$-local representations.

math.RT↗
arXiv2026

On Universal Virtual and Welded Braid Groups and Their Linear Representations

We introduce linear representations of the universal virtual braid group $UV_n(c)$, where $n\geq 2$ and $c\geq 1$, which is a unifying framework for braid-type groups with multiple types of crossings. We classify and study its complex homogeneous $2$-local representations for all $n\geq 3$ and $c\geq 1$ (unique up to equivalence) and complex homogeneous $3$-local representations for all $n\geq 4$ and $c=2$ (four distinct families). We then introduce the universal welded braid group $UW_n(c)$ as a quotient of $UV_n(c)$ by the welded relations. This group recovers all known welded-type groups as quotients. We prove that $UW_n(c)$ has abelianization $\mathbb{Z}^c \oplus \mathbb{Z}_2$, perfect commutator subgroup for $n \geq 5$, trivial center, and $S_n$ as its smallest non-abelian finite quotient. Finally, we classify and study the complex homogeneous $2$-local representations of $UW_n(c)$ for all $n\geq 3$ and $c\geq 1$, obtaining three distinct families.

math.RT↗
arXiv2026

On virtual singular braid groups

The virtual singular braid group arises as a natural common generalization of classical singular braid groups and virtual braid groups. In this paper, we study several algebraic properties of the virtual singular braid group $VSG_n$. We introduce numerical invariants for virtual singular braids arising from exponent sums of words in $VSG_n$, and describe explicitly the kernels of the associated homomorphisms onto abelian groups. We then determine all group homomorphisms, up to conjugation, from $VSG_n$ to the symmetric group $S_n$, and obtain corresponding semi-direct product decompositions. In the particular case $n=2$, we provide explicit presentations and algebraic descriptions of the kernels. Moreover, we show that certain relations are forbidden in $VSG_n$, and we introduce and study natural quotients of the virtual singular braid group, including welded and unrestricted versions, for which analogous structural results are obtained.

math.GR↗
arXiv2026

Positive cones and bi-orderings on almost-direct products of free groups

Almost-direct products of free groups arise naturally in braid theory and in the study of automorphism groups of free groups. Although bi-invariant orderings are known to exist for many such groups, their explicit structure is often left implicit. In this paper, we give an explicit description of the positive cones defining bi-invariant orderings on almost-direct products of free groups, using normal forms derived from the almost-direct product decomposition together with Magnus-type orderings on free factors. We establish key structural properties of these cones, including compatibility with natural projections, convexity of canonical subgroups, and invariance under suitable classes of automorphisms. As applications, we show how the construction applies to several families of groups of geometric and algebraic interest, such as pure monomial braid groups and McCool groups.

math.GR↗
arXiv2026

Universal virtual braid groups

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 $n!$.

math.GR↗
arXiv2026

Permutational wreath pullbacks and framed braid-type groups

Let $σ\colon G \to S_n$ be a surjective homomorphism and let $H$ be a group. We introduce the \emph{permutational wreath pullback} \[ H \wr_σG = H^n \rtimes_σG, \] where the action of $G$ on $H^n$ is induced by permutation of coordinates via $σ$, and undertake a systematic structural study of this construction. We determine the center and the abelianization in full generality. We further show that $H \wr_σG$ admits a natural interpretation as the pullback of the classical wreath product $H \wr S_n$ along $σ$, providing a conceptual explanation for its functorial behavior. When $H$ is finitely generated abelian, we establish a criterion for the abelian kernel $H^n$ to be characteristic and for $H \wr_σG$ to inherit the $R_\infty$-property from $G$; we verify this criterion for kernels arising from the virtual braid group $VB_n$ and the virtual twin group $VT_n$, obtaining new families of framed groups with the $R_\infty$-property. Rigidity results show that the abelian kernel, $n$, $H$, and $G$ are determined by the abstract group $H \wr_σG$. Applications include uniform descriptions of classical, surface, virtual, and singular framed braid groups, and a reduction of splitting problems for framed surface braid groups to the classical Fadell--Neuwirth setting.

math.GR↗
arXiv2025

Powers of half-twists and congruence subgroups of braid groups

In this work, we study the relationship between congruence subgroups $B_n[m]$ and $\mathcal{N}_n(σ_1^m)$ the normal closure of $σ_1^m$, where $σ_1$ is the classical generator of $B_n$. We characterize the conditions under which $\mathcal{N}_n(σ_1^m)$ has finite index in $B_n[m]$ and provide explicit generators for these finite quotients. For the cases where the index is infinite, we show that $B_n[m]/\mathcal{N}_n(σ_1^m)$ contains a free subgroup. Additionally, we compute the Abelianisation of Coxeter braid subgroups in the finite index cases and construct new finite quotients using commutators of congruence subgroups.

math.GR↗
arXiv2025

The R$_{\infty}$-property for braid groups over orientable surfaces

Let $Σ_{g,p}$ be an orientable surface of genus $g$ and of finite type without boundary (i.e. an orientable closed surface with a finite number $p$ of points removed). In this paper we study the R$_{\infty}$-property for the surface pure braid groups $P_n(Σ_{g,p})$ as well as for the full surface braid groups $B_n(Σ_{g,p})$. We show that, with few exceptions, these groups have the R$_{\infty}$-property.

math.GT↗
arXiv2024

Skew-symmetric augmented matrices and a characterization of virtual doodles

In this paper, we present a brief overview of the concept of doodles from the perspective of J.S. Carter's work on classifying immersed curves and the work of J.S. Carter, S. Kamada, and M. Saito on stable equivalence of knots on surfaces and virtual knot cobordisms. We use the homology intersection number and the work of G. Cairns and D. Elton on the Gauss word problem to introduce the concept of skew-symmetric augmented matrices for determining whether a virtual doodle is non-classical. We also provide a characterization of the virtualization of classical doodles.

math.GT↗
arXiv2024

Coxeter-type quotients of surface braid groups

Let $M$ be a closed surface, $q\geq 2$ and $n\geq 2$. In this paper, we analyze the Coxeter-type quotient group $B_n(M)(q)$ of the surface braid group $B_{n}(M)$ by the normal closure of the element $σ_1^q$, where $σ_1$ is the classic Artin generator of the Artin braid group $B_n$. Also, we study the Coxeter-type quotient groups obtained by taking the quotient of $B_n(M)$ by the commutator subgroup of the respective pure braid group $[P_n(M),P_n(M)]$ and adding the relation $σ_1^q=1$, when $M$ is a closed orientable surface or the disk.

math.GR↗
arXiv2024

Congruence subgroups of braid groups and crystallographic quotients. Part I

This paper is the first of a two part series devoted to describing relations between congruence and crystallographic braid groups. We recall and introduce some elements belonging to congruence braid groups and we establish some (iso)-morphisms between crystallographic braid groups and corresponding quotients of congruence braid groups.

math.GR↗
arXiv2024

Characteristic subgroups and the R$_\infty$-property for virtual braid groups

Let $n\geq 2$. Let $VB_n$ (resp. $VP_n$) denote the virtual braid group (resp. virtual pure braid group), let $WB_n$ (resp. $WP_n$) denote the welded braid group (resp. welded pure braid group) and let $UVB_n$ (resp. $UVP_n$) denote the unrestricted virtual braid group (resp. unrestricted virtual pure braid group). In the first part of this paper we prove that, for $n\geq 4$, the group $VP_n$ and for $n\geq 3$ the groups $WP_n$ and $UVP_n$ are characteristic subgroups of $VB_n$, $WB_n$ and $UVB_n$, respectively. In the second part of the paper we show that, for $n\geq 2$, the virtual braid group $VB_n$, the unrestricted virtual pure braid group $UVP_n$, and the unrestricted virtual braid group $UVB_n$ have the R$_\infty$-property. As a consequence of the technique used for few strings we also prove that, for $n=2,3,4$, the welded braid group $WB_n$ has the R$_\infty$-property and that for $n=2$ the corresponding pure braid groups have the R$_\infty$-property. On the other hand for $n\geq 3$ it is unknown if the R$_\infty$-property holds or not for the virtual pure braid group $VP_n$ and the welded pure braid group $WP_n$.

math.GR↗
arXiv2023

Virtual braid groups, virtual twin groups and crystallographic groups

Let $n\ge 2$. Let $VB_n$ (resp. $VP_n$) be the virtual braid group (resp. the pure virtual braid group), and let $VT_n$ (resp. $PVT_n$) be the virtual twin group (resp. the pure virtual twin group). Let $Π$ be one of the following quotients: $VB_n/Γ_2(VP_n)$ or $VT_n/Γ_2(PVT_n)$ where $Γ_2(H)$ is the commutator subgroup of $H$. In this paper, we show that $Π$ is a crystallographic group and we characterize the elements of finite order and the conjugacy classes of elements in $Π$. Furthermore, we realize explicitly some Bieberbach groups and infinite virtually cyclic groups in $Π$. Finally, we also study other braid-like groups (welded, unrestricted, flat virtual, flat welded and Gauss virtual braid group) module the respective commutator subgroup in each case.

math.GR↗
arXiv2022

Seifert circles, crossing number and the braid index of generalized knots and links

For classical links Ohyama proved an inequality involving the minimal crossing number and the braid index, then motivated from this Takeda showed an analogous inequality for virtual links. In this paper, we are interested in studying properties of links independent of the type of crossings, and for this reason, we introduce generalized crossings for diagrams and generalized Reidemeister-type moves. The aim of this work is to prove the same type of inequality mentioned above but now involving the total crossing number and the braid index of generalized knots and links. In particular, we show that the result holds for virtual singular links.

math.GT↗
arXiv2021

The conjugacy problem and virtually cyclic subgroups in the Artin braid group quotient $B_n/[P_n,P_n]$

Let $n\geq 3$. In this paper we deal with the conjugacy problem in the Artin braid group quotient $B_n/[P_n,P_n]$. To solve it we use systems of equations over the integers arising from the action of $B_n/[P_n,P_n]$ over the abelianization of the pure Artin braid group $P_n/[P_n,P_n]$. Using this technique we also realize explicitly infinite virtually cyclic subgroups in $B_n/[P_n,P_n]$.

math.GR↗