arXiv research

Study on braid groups' congruence subgroups and their crystallographic quotients.

Paper explores relations between braid groups and their quotients.

Sharp bounds found on nonabelian quotients of surface braid groups.

We solved a conjecture about braid group quotients being alternating groups.

Study finds smallest non-trivial quotients of braid groups and commutator subgroups.

The paper studies Coxeter quotients of surface braid groups.

This paper classifies all admissible quotients of a surface braid group up to order 127.

We derive a lower bound on the size of finite non-cyclic quotients of the braid group that is superexponential in the number of strands. We also derive a similar lower bound for nontrivial finite quotients of the commutator subgroup of the braid group.

The study explores congruence subgroups of braid groups and their quotients.

New groups from surface braids help create complex geometric shapes.

A new presentation of a quotient of braid groups leads to a new type of Burnside group.

The paper solves the conjugacy problem in a specific braid group quotient and finds infinite virtually cyclic subgroups.

Study on braid group quotients by congruence subgroups.

New proofs show smallest non-cyclic quotients for braid and mapping class groups.

New insights into a complex hyperbolic braid group quotient.

Study of Dehn filling quotients in hierarchically hyperbolic groups.

The study characterizes subgroups of braid groups and their finite quotients.

Unified algebraic framework for virtual braid structures with strong structural consequences.

We consider the group of unrestricted virtual braids, describe its structure and explore its relations with fused links. Also, we define the groups of flat virtual braids and virtual Gauss braids and study some of their properties, in particular their linearity.

Study the relationship between orbit braid group and equivariant mapping class group on surfaces.

We study a quotient of the group algebra of the braid group in which the Artin generators satisfy a cubic relation. This quotient is maximal among the ones satisfying such a cubic relation. It is finite-dimensional for at least n at most 5 and we investigate its module structure in this range. We also investigate the p…

This work shows non-abelian quotient groups in string link concordance.

The study examines subgroups of braid groups related to symmetric groups.

Study on virtual singular braid groups with algebraic properties and homomorphisms.

We give generators for a certain complex hyperbolic braid group. That is, we remove a hyperplane arrangement from complex hyperbolic $13$-space, take the quotient of the remaining space by a discrete group, and find generators for the orbifold fundamental group of the quotient. These generators have the most natural fo…

Study extra-special quotients of surface braid groups and construct double Kodaira fibrations.

We define the braid groups of a two-dimensional orbifold and introduce conventions for drawing braid pictures. We use these to realize the Artin groups associated to the spherical Coxeter diagrams A_n, B_n=C_n and D_n and the affine diagrams tilde{A}_n, tilde{B}_n, tilde{C}_n and tilde{D}_n as subgroups of the braid gr…

We study the structure of the virtual braid group. It is shown that the virtual braid group is a semi--direct product of the virtual pure braid group and the symmetric group. Also, it is shown that the virtual pure braid group is a semi--direct product of free groups. From these results we obtain a normal form of words…

We prove that the quotients of the group algebra of the braid group on 3 strands by a generic quartic and quintic relation respectively, have finite rank. This is a special case of a conjecture by Broué, Malle and Rouquier for the generic Hecke algebra of an arbitrary complex reflection group. Exploring the consequence…

We prove that the braid group $B_4$ on 4 strings, as well as its central quotient $B_4/< z>$, have the property RD of Haagerup-Jolissaint. It follows that the automorphism group $\Aut(F_2)$ of the free group $F_2$ on 2 generators has property RD. We also prove that the braid group $B_4$ is a group of intermediate rank …

Following an idea of Gonçalvez, Guaschi and Ocampo on the usual braid group we construct crystallographic and Bieberbach groups as (sub)quotients of the generalized braid group associated to an arbitrary complex reflection group.

The abstract discusses braided surfaces and their characteristic maps, linking them to algebraic and geometric properties.

Homotopy braid groups are shown to be torsion-free.

The aim of the present note is to show that the natural map from classical braids to virtual braids is an inclusion; this proof does not use any complete invariants of classical braids; it is based on the projection from virutal braids to classical braids (similar to the one given in \cite{Projection}); this projection…

We prove that the quotients of the group algebra of the braid group introduced by L. Funar in Comm. Math. Phys., 1995, collapses in characteristic distinct from 2. In characteristic 2 we define several quotients of it, which are connected to the classical Hecke and Birman-Wenzl-Murakami quotients, but which admit in ad…

In this paper we calculate fundamental groups (and some of their quotients) of complements of four toric varieties branch curves. For these calculations, we study properties and degenerations of these toric varieties and the braid monodromies of the branch curves in $\mathbb{CP}^2$. The fundamental groups related to th…

The study explores splitting conditions for mixed braid group sequences.

In this paper we define the $p$-adic framed braid group ${\mathcal F}_{\infty,n}$, arising as the inverse limit of the modular framed braids and we give topological generators for ${\mathcal F}_{\infty, n}$. We also give geometric interpretations for the $p$-adic framed braids. We then construct a $p$-adic Yokonuma-Hec…

The braid group of a complex reflection group is shown to be an index d subgroup.

Spider category comparison proves equivalence to Sikora's quotient category.

Let $\widetilde{K}$ be a two-periodic braid and let $K$ be its quotient. In this paper we show there is a spectral sequence from the next-to-top winding number grading of the sutured annular Khovanov homology of the closure of $\widetilde{K}$ to the next-to-top winding number grading of the sutured annular Khovanov hom…

We interpret Coxeter's truncated braid groups in terms of Platonic solids.

Study of generalized J-groups and their presentations.

The paper characterizes crystallographic groups derived from virtual braid and twin groups.

The purpose of this article is to describe connections between the loop space of the 2-sphere, Artin's braid groups, a choice of simplicial group whose homotopy groups are given by modules called Lie(n), as well as work of Milnor, and Habegger-Lin on "homotopy string links". The current article exploits Lie algebras as…

This paper aims to generalize Artin's ideas to establish an one-to-one correspondence between the orbit braid group $B^{orb}_n(\mathbb{C},\mathbb{Z}_p)$ and a quotient of a group formed by some particular homeomorphisms of a punctured plane. First, we find a faithful representation of $B^{orb}_n(\mathbb{C},\mathbb{Z}_p…

This paper presents a new exact sequence for orbifold braid groups and mapping class groups.

This article was submitted to a volume under preparation, with Benson Farb as the editor, on the topic of open problems in surface mapping class groups. The braid group B_n is the mapping class group of an n-times punctured disk. The Iwahori-Hecke algebra H_n is a quotient of the braid group algebra of B_n by a quadrat…