A Special Subgroup of the Surface Braid Group

Conjugacy problem for subgroups with applications to Artin groups and braid type groups

Let $G$ be a group endowed with a solution to the conjugacy problem and with an algorithm which computes the centralizer in $G$ of any element of $G$. Let $H$ be a subgroup of $G$. We give some conditions on $H$, under which we provide a solution to the conjugacy problem in $H$. We apply our results to some Artin groups and braid type groups. In particular, we give explicit solutions to the conjugacy problem in the Artin groups of type $\tilde{A}_n$ and $\tilde{C}_n$.

Gassner表示的纯辫子群像具有成对自由生成元

The Image of the Gassner Representation of the Pure Braid Subgroup has Pairwise Free Generators

本文研究Gassner表示在纯辫子群上的像，通过Colored-Burau表示等价性，利用Lipschutz方法结合线性代数分析，证明其具有成对自由生成元的性质。

曲面辫子群与虚拟辫子群的子群可分性

Subgroup separability of surface braid groups and virtual braid groups

该工作考察曲面辫子群与虚拟（奇异）辫子群的子群可分性（LERF）及相关性质。

纯无限辫子Thompson群

Pure infinitely braided Thompson groups

该研究证明无限辫子Thompson群的纯子群具有双可序性，并为所有有限生成纯子群提供了显式生成元集，推进了群论相关研究。

A presentation for Hilden's subgroup of the braid group

Consider the unit ball, B = D x [0,1], containing n unknotted arcs a_1, ... a_n such that the boundary of each a_i lies in D x {0}. We give a finite presentation for the mapping class group of B fixing the arcs {a_1, ..., a_n} setwise and fixing D x {1} pointwise. This presentation is calculated using the action of this group on a simply-connected complex.

使用辫群中的米哈伊洛娃子群进行密钥交换构造

A Key Exchange Construction using Mihailova Subgroups in Braid groups

研究利用辫群中米哈伊洛娃子群进行密钥交换构造，源于其成员问题不可解，引出商群公式，但代数事实不能提供完整安全证明，此构造是有代数动机的候选方案，完整密码安全性待进一步研究。

Orbifold辫群与复辫群

Orbifold braid groups and complex braid groups

本文将 Allock 关于含至多两个二阶锥点的 orbifold 辫群包含特定 Artin 群有限指数子群的结果推广至任意阶锥点，并证明这些子群在许多情况下可识别为复辫群。

Permutation representations of the braid group commutator subgroup

We study the representations of the commutator subgroup of the braid group with n strands in the symmetric group of degree r. Motivated by some experimental results, we conjecture that for n>r, every such representation is trivial.

Double Centralizers of Parabolic Subgroups of Braid Groups

We characterize the double centralizer of all parabolic subgroups of the braid groups. We apply this result to provide a new and potentially more efficient solution to the subgroup conjugacy problem for parabolic subgroups. In the course of the proof we also characterize the centralizer for all parabolic subgroups.

The congruence subgroup problem for braid groups: Thurston's proof

In this article we present an unpublished proof of W. Thurston that pure braid groups have the congruence subgroup property.

Torsion subgroups of quasi-abelianized braid groups

This article extends the works of Gonçalves, Guaschi, Ocampo [GGO] and Marin [MAR2] on finite subgroups of the quotients of generalized braid groups by the derived subgroup of their pure braid group. We get explicit criteria for subgroups of the (complex) reflection group to lift to subgroups of this quotient. In the specific case of the classical braid group, this enables us to describe all its finite subgroups : we show that every odd-order finite group can be embedded in it, when the number of strands goes to infinity. We also determine a complete list of the irreducible reflection groups f…

辫群的同余子群与晶体学商群. 第二部分

Congruence subgroups of braid groups and crystallographic quotients. Part II

本文接续前期研究，针对与辫群$B_3$相关的同余辫群下中心列展开分析，证明其对应商群具有几乎晶体学性质，推进了同余子群与晶体学辫群领域的研究。

Homomorphisms of commutator subgroups of braid groups

We give a complete classification of homomorphisms from the commutator subgroup of the braid group on $n$ strands to the braid group on $n$ strands when $n$ is at least 7. In particular, we show that each nontrivial homomorphism extends to an automorphism of the braid group on $n$ strands. This answers four questions of Vladimir Lin. Our main new tool is the theory of totally symmetric sets.

Smallest nontrivial quotients of the commutator subgroup of braid groups

We prove that the smallest non-trivial quotients of the commutator subgroups of the braid groups are the alternating groups, proving a conjecture of Chudnovsky-Kordek-Li-Partin. Furthermore, we show that any minimal quotient map is the standard projection, composed with an automorphism of the alternating group.

Classifying spaces for the family of virtually cyclic subgroups of braid groups

We prove that, for $n\geq 3$, the minimal dimension of a model of the classifying space of the full braid group $B_n$, and of the pure braid group $P_n$, with respect to the family of virtually cyclic groups is $n$.

Densely ordered braid subgroups

Dehornoy showed that the Artin braid groups $B_n$ are left-orderable. This ordering is discrete, but we show that, for $n >2$ the Dehornoy ordering, when restricted to certain natural subgroups, becomes a dense ordering. Among subgroups which arise are the commutator subgroup and the kernel of the Burau representation (for those $n$ for which the kernel is nontrivial). These results follow from a characterization of least positive elements of any normal subgroup of $B_n$ which is discretely ordered by the Dehornoy ordering.

Conjugation-invariant norms on the commutator subgroup of the infinite braid group

In this paper, we give a proof of the result of Brandenbursky and Kȩdra which says that the commutator subgroup of the infinite braid group admits stably unbounded norms. Moreover, we observe the norms which we constructed are equivalent to the biinvariant word norm studied by Brandenbursky and Kȩdra.

Two-generator subgroups of the pure braid group

We show that any two elements of the pure braid group either commute or generate a free group, settling a question of Luis Paris. Our proof involves the theory of 3-manifolds and the theory of group actions on trees.

Commutator Subgroups of Welded Braid Groups

Let $WB_n$ be the welded (or loop) braid group on n strands, $n \geq 3$. We investigate commutator subgroup of $WB_n$. We prove that the commutator subgroup $WB_n'$ is finitely generated and Hopfian. We show that $WB_n'$ is perfect if and only if $n \geq 5$. We also compute finite presentation for $FWB_n'$, the commutator subgroup of the flat welded braid group $FWB_n$. Along the way, we investigate adorability of these groups.

The commutator subgroup of the braid group is generated by two elements

For $n$ at least 7 and $n$ equal to 5, we give generating sets of size 2 for the commutator subgroup of the braid group on $n$ strands. These generating sets are of the smallest possible cardinality. For $n$ equal to 4 or 6, we give generating sets of size three. We also prove that the commutator subgroup of the braid group on 4 strands cannot be generated by fewer than three elements.

Embeddings of the braid groups of covering spaces, classification of the finite subgroups of the braid groups of the real projective plane, and linearity of braid groups of low-genus surfaces

Let M be a compact, connected surface, possibly with a finite set of points removed from its interior. Let d,n be positive integers, and let N be a d-fold covering space of M. We show that the covering map induces an embedding of the n-th braid group B_n(M) of M in the (dn)-th braid group B_{dn}(N) of N, and give several applications of this result. First, we classify the finite subgroups of the n-th braid group of the real projective plane, from which we deduce an alternative proof of the classification of the finite subgroups of the mapping class group of the n-punctured real projective plan…

On quotients of congruence subgroups of braid groups

The integral Burau representation provides a map from the braid group into a group of integral matrices. This allows for a definition of congruence subgroups of the braid group as the preimage of the usual principal congruence subgroups of integral matrices. We explore the structure these congruence subgroups by examining some of the quotients that may arise in the series induced by divisibility of levels. We build on the work of Stylianakis on symmetric quotients of congruence subgroups, which itself generalizes the quotient of the braid group by the pure braid group. We accomplish this by ut…

辫群的同余子群与晶体学商群。第一部分

Congruence subgroups of braid groups and crystallographic quotients. Part I

本文为系列第一部分，通过引入同余辫群元素，建立了晶体学辫群与同余辫群商群间的同构关系。

特征子群与虚拟辫群的R$_\infty$性质

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

本文证明了虚拟纯辫群、焊接纯辫群等在特定阶数下为相应辫群的特征子群，并证明虚拟辫群、无限制虚拟辫群等具有R$_\infty$性质，同时指出部分纯辫群在特定阶数下的该性质仍属未知。

Alternating subalgebras of Hecke algebras and alternating subgroups of braid groups

For a Coxeter system (G,S) the multi-parametric alternating subalgebra H^+(G) of the Hecke algebra and the alternating subgroup B^+(G) of the braid group are defined. Two presentations for H^+(G) and B^+(G) are given; one generalizes the Bourbaki presentation for the alternating subgroups of Coxeter groups, another one uses generators related to edges of the Coxeter graph.

Singular braids, singular links and subgroups of camomile type

In this paper we find a finite set of generators and defining relations for the singular pure braid group $SP_n$, $n \geq 3$, that is a subgroup of the singular braid group $SG_n$. Using this presentation, we prove that the center of $SG_n$ (which is equal to the center of $SP_n$ for $n \geq 3$) is a direct factor in $SP_n$ but it is not a direct factor in $SP_n$. We introduce subgroups of camomile type and prove that the singular pure braid group $SP_n$, $n \geq 5$, is a subgroup of camomile type in $SG_n$. Also we construct the fundamental singquandle using a representation of the singular b…

由Birman-Ko-Lee生成元生成的辫群子群

Subgroups of braid groups generated by Birman-Ko-Lee generators

本文研究了由Birman-Ko-Lee生成元生成的辫群子群，提出了一种判断成员资格的标准和将元素表示为生成元乘积的算法，基于图示方法分析Hurwitz作用。

On symmetric commutator subgroups, braids, links and homotopy groups

In this paper, we investigate some applications of commutator subgroups to homotopy groups and geometric groups. In particular, we show that the intersection subgroups of some canonical subgroups in certain link groups modulo their symmetric commutator subgroups are isomorphic to the (higher) homotopy groups. This gives a connection between links and homotopy groups. Similar results hold for braid and surface groups.

The quaternion group as a subgroup of the sphere braid groups

Let n be greater than or equal to 3. We prove that the quaternion group of order 8 is realised as a subgroup of the sphere braid group B\_n(S^2) if and only if n is even. If n is divisible by 4 then the commutator subgroup of B\_n(S^2) contains such a subgroup. Further, for all n greater than or equal to 3, B\_n(S^2) contains a subgroup isomorphic to the dicyclic group of order 4n.

Representations of braid groups via conjugation actions on congruence subgroups

We construct two families of representations of the braid group $B_n$ by considering conjugation actions on congruence subgroups of $GL_{n-1}(Z[t^{\pm 1},q^{\pm 1}])$. We show that many of these representations are faithful modulo the center of $B_n$.

A virtual geometric action of a braid group

We study a geometric action on a CAT(0) space of a finite index subgroup of the quotient group of the braid group on 4 strands by its center.

Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane

We classify the (finite and infinite) virtually cyclic subgroups of the pure braid groups $P_{n}(RP^2)$ of the projective plane. The maximal finite subgroups of $P_{n}(RP^2)$ are isomorphic to the quaternion group of order 8 if $n=3$, and to $\Z_{4}$ if $n\geq 4$. Further, for all $n\geq 3$, up to isomorphism, the following groups are the infinite virtually cyclic subgroups of $P_{n}(RP^2)$: $\Z$, $\Z_{2} \times \Z$ and the amalgamated product $\Z_{4} \ast_{\Z_{2}} \Z_{4}$.

Commutator Subgroups of Singular Braid Groups

The singular braids with $n$ strands, $n \geq 3$, were introduced independently by Baez and Birman. It is known that the monoid formed by the singular braids is embedded in a group that is known as singular braid group, denoted by $SG_n$. There has been another generalization of braid groups, denoted by $GVB_n$, $n \geq 3$, which was introduced by Fang as a group of symmetries behind quantum quasi-shuffle structures. The group $GVB_n$ simultaneously generalizes the classical braid group, as well as the virtual braid group on $n$ strands. We investigate the commutator subgroups $SG_n'$ and $GVB…

Anti-trees and right-angled Artin subgroups of braid groups

We prove that an arbitrary right-angled Artin group $G$ admits a quasi-isometric group embedding into a right-angled Artin group defined by the opposite graph of a tree. Consequently, $G$ admits quasi-isometric group embeddings into a pure braid group and into the area-preserving diffeomorphism groups of the 2--disk and the 2--sphere, answering questions due to Crisp--Wiest and M. Kapovich. Another corollary is that a pure braid group contains a closed hyperbolic manifold group as a quasi-isometrically embedded subgroup up to dimension eight. Finally, we show that the isomorphism problem, conj…

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 $B_n/[P_n,P_n]$.

On the Andreadakis equality for a subgroup of the McCool group

The McCool group has families of subgroups such as the ordinary pure braid group, the upper triangular McCool group and the partial inner automorphism group. The generalized Andreadakis conjecture holds for the ordinary pure braid group and the upper triangular McCool group. In this paper, we prove a similar result for the partial inner automorphism group and also establish an isomorphism between this group and the inner automorphism group of the upper triangular McCool group.

少股数辫子群的换位子群的同态

Homomorphisms of commutator subgroups of braid groups with small number of strings

本文刻画任意股数辫子群 $B_n$ 及其换位子群 $B'_n$ 的自同态与相关同态，补足小股数情形，并对 $n=4$ 与 $n\ge5$ 使用不同方法。

Normal subgroups of the braid group and the metaconjecture of Ivanov

We show that many normal subgroups of the braid group modulo its centre, and of the mapping class group of a sphere with marked points, have the property that their automorphism and abstract commensurator groups are mapping class groups of such spheres. As one application, we establish the automorphism groups of each term in the lower central series and derived series of the pure braid group. We also obtain new proofs of results of Dyer-Grossman and Orevkov, showing that the automorphism groups of the braid group and of its commutator subgroup are isomorphic. We then calculate the automorphism…

Brunnian Braids and Lie Algebras

Brunnian braids have interesting relations with homotopy groups of spheres. In this work, we study the graded Lie algebra of the descending central series related to Brunnian subgroup of the pure braid group. A presentation of this Lie algebra is obtained.

辫群模其同余子群的商群

Quotients of braid groups by their congruence subgroups

该研究解决了Dan Margalit提出的公开问题，确定了辫群同余子群在整一般线性群中的像，并利用辛同余子群刻画了辫群模同余子群的商群结构。

Presentations of subgroups of the braid group generated by powers of band generators

According to the Tits conjecture proved by Crisp and Paris, [CP], the subgroups of the braid group generated by proper powers of the Artin elements are presented by the commutators of generators which are powers of commuting elements. Hence they are naturally presented as right-angled Artin groups. The case of subgroups generated by powers of the band generators is more involved. We show that the groups are right-angled Artin groups again, if all generators are proper powers with exponent at least 3. We also give a presentation in cases at the other extreme, when all generators occur with expo…

Quasi-isometrically embedded subgroups of braid and diffeomorphism groups

We show that a large class of right-angled Artin groups (in particular, those with planar complementary defining graph) can be embedded quasi-isometrically in pure braid groups and in the group of area preserving diffeomorphisms of the disk fixing the boundary (with respect to the $L^2$-norm metric); this extends results of Benaim and Gambaudo who gave quasi-isometric embeddings of $F\_n$ and $\Z^n$ for all $n>0$. As a consequence we are also able to embed a variety of Gromov hyperbolic groups quasi-isometrically in pure braid groups and in the diffeomorphism group of the disk. Examples includ…

辫群水平 $m$ 同余子群的生成元

Generators for the level $m$ congruence subgroups of braid groups

本文证明辫群水平 $m$ 同余子群 $B_n[m]$ 由半扭转的 $m$ 次幂与辫 Torelli 群生成，解决 Margalit 问题并推广已有结果。

复辫子群的抛物子群：剩余情形

Parabolic subgroups of complex braid groups: the remaining case

本文利用复辫子群 B(G31) 的 Garside 广群结构，证明 Marin 和 González-Meneses 关于抛物子群的主要定理，补全了此前唯一遗留情形。

Braid groups of normalizers of reflection subgroups

Let $W_0$ be a reflection subgroup of a finite complex reflection group $W$, and let $B_0$ and $B$ be their respective braid groups. In order to construct a Hecke algebra $\widetilde{H}_0$ for the normalizer $N_W(W_0)$, one first considers a natural subquotient $\widetilde{B}_0$ of $B$ which is an extension of $N_W(W_0)/W_0$ by $B_0$. We prove that this extension is split when $W$ is a Coxeter group, and deduce a standard basis for the Hecke algebra $\widetilde{H}_0$. We also give classes of both split and non-split examples in the non-Coxeter case.

Commutator Subgroups of Virtual and Welded Braid Groups

Let $VB_n$, resp. $WB_n$ denote the virtual, resp. welded, braid group on $n$ strands. We study their commutator subgroups $VB_n' = [VB_n, VB_n]$ and, $WB_n' = [WB_n, WB_n]$ respectively. We obtain a set of generators and defining relations for these commutator subgroups. In particular, we prove that $VB_n'$ is finitely generated if and only if $n \geq 4$, and $WB_n'$ is finitely generated for $n \geq 3$. Also we prove that $VB_3'/VB_3'' =\mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus\mathbb{Z}_3 \oplus \mathbb{Z}^{\infty}$, $VB_4' / VB_4'' = \mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_3$, $WB…

The center of pure complex braid groups

Broué, Malle and Rouquier conjectured in that the center of the pure braid group of an irreducible finite complex reflection group is cyclic. We prove this conjecture, for the remaining exceptional types, using the analogous result for the full braid group due to Bessis, and we actually prove the stronger statement that any finite index subgroup of such braid group has cyclic center.

The classification and the conjugacy classes of the finite subgroups of the sphere braid groups

Let n\geq 3. We classify the finite groups which are realised as subgroups of the sphere braid group B_n(S^2). Such groups must be of cohomological period 2 or 4. Depending on the value of n, we show that the following are the maximal finite subgroups of B_n(S^2): Z_{2(n-1)}; the dicyclic groups of order 4n and 4(n-2); the binary tetrahedral group T_1; the binary octahedral group O_1; and the binary icosahedral group I. We give geometric as well as some explicit algebraic constructions of these groups in B_n(S^2), and determine the number of conjugacy classes of such finite subgroups. We also …

扭曲虚拟辫子群

Twisted Virtual Braid Group

本文研究扭曲虚拟辫子群TVBₙ的子群结构与半直积分解，确定了TVPₙ、TVHₙ等子群的生成元、定义关系及分解形式，得出VPₙ是TVPₙ子群的结论，丰富了虚拟辫子群的代数结构研究。

Cohomology of braids, principal congruence subgroups and geometric representations

The main purpose of this article is to give the integral cohomology of classical principal congruence subgroups in SL(2,Z) as well as their analogues in the third braid group with local coefficients in symmetric powers of the natural symplectic representation. The resulting answers (1) correspond to certain modular forms in characteristic zero, and (2) the cohomology of certain spaces in homotopy theory in characteristic p. The torsion is given in terms of the structure of a "p-divided power algebra". The work is an extension of the work in arXiv:1204.5390v1 as well as extensions of a classica…

复辫子群的抛物子群

Parabolic subgroups of complex braid groups

本文针对与任意不可约复反射群相关的广义辫子群引入抛物子群类，证明其构成格且每个元素有抛物闭包，还推广了曲线复形并猜想其双曲性。

虚拟奇异辫子之间的代数结构

Algebraic structures among virtual singular braids

该研究证明n股虚拟奇异辫子幺半群可嵌入群VSGₙ，给出其半直积结构与VSPGₙ的表示及子群分解，为虚拟奇异辫子的代数结构分析提供基础。

半扭转的幂与辫群的同余子群

Powers of half-twists and congruence subgroups of braid groups

本文研究辫群同余子群$B_n[m]$与生成元幂正规闭包的关系，刻画前者含后者有限指数的条件并给出有限商群生成元，对无限指数情况证明商群含自由子群，还计算了相关交换化并构造新有限商群。

Metric properties of the braided Thompson's groups

Braided Thompson's groups are finitely presented groups introduced by Brin and Dehornoy which contain the ordinary braid groups $B_n$, the finitary braid group $B_{\infty}$ and Thompson's group $F$ as subgroups. We describe some of the metric properties of braided Thompson's groups and give upper and lower bounds for word length in terms of the number of strands and the number of crossings in the diagrams used to represent elements.