Two-generator subgroups of the pure braid group
Christopher J Leininger, Dan Margalit
我们证明了纯辫群的任意两个元素要么交换要么生成一个自由群，解决了Luis Paris的一个问题。我们的证明涉及到三维流形理论和树上的群作用理论。
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.
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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。
The congruence subgroup problem for braid groups: Thurston's proof
在这篇文章中，我们提出了一个未发表的证明W。Thurston证明了纯辫群具有同余子群性质.
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。
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.
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.
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.
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.
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
设M是一个紧致的连通曲面，可能从其内部移去一个有限集点。设d，n是正整数，N是M的d重覆盖空间.证明了覆叠映射诱导M的第n个辫群B_n（M）嵌入N的第（dn）个辫群B_{dn}（N），并给出了这一结果的几个应用.首先，我们对实射影平面的n辫群的有限子群进行了分类，由此导出了Bujalance，Cirre和Gamboa关于n-穿孔实射影平面的映射类群的有限子群分类的另一种证明.其次，利用Bigelow和Krammer的B_{n}的线性性，我们证明了低亏格的紧致连通曲面的辫群是线性的。
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…
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。
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…
The quaternion group as a subgroup of the sphere braid groups
设n大于或等于3。证明了8阶四元数群可实现为球面辫群B n（S^2）的子群当且仅当n是偶数.若n能被4整除，则Bn（S^2）的换位子群包含这样一个子群.进一步地，对所有大于或等于3的n，Bn（S^2）包含同构于4n阶双循环群的子群.
Classification of the virtually cyclic subgroups of the pure braid groups of the projective plane
对射影平面上纯辫群$P_{n}(RP^2)$的（有限和无限）虚循环子群进行了分类。$P_{n}(RP^2)$的极大有限子群同构于8阶四元数群，若$n=3$，且同构于$\Z_{4}$，若$n\geq 4$。此外，对于所有的$n\geq 3$，直到同构，以下群是$P_{n}(RP^2)$的无限虚循环子群：$\Z$，$\Z_{2} \times \Z$和合并积$\Z_{4} \ast_{\Z_{2}} \Z_{4}$。
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…
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…
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 $B_n/[P_n,P_n]$.
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…
Presentations of subgroups of the braid group generated by powers of band generators
根据Crisp和Paris证明的Tits猜想[CP]，由Artin元的真幂生成的辫群的子群由交换元的幂生成元的生成子表示.因此，它们自然地被表示为直角Artin群。由带生成元的幂生成的子群的情况更为复杂。我们证明了，如果所有的生成元都是指数至少为3的真幂，则这些群又是直角Artin群。我们还介绍了在另一个极端的情况下，当所有的发电机发生指数为1或2，这是远远不是直角阿廷组。
On symmetric commutator subgroups, braids, links and homotopy groups
本文研究了换位子群在同伦群和几何群中的一些应用。特别地，我们证明了某些环群中某些典型子群的交子群模其对称换位子群同构于（高）同伦群。这给出了链路和同伦群之间的联系。编织物和表面组的结果相似。
Quasi-isometrically embedded subgroups of braid and diffeomorphism groups
我们表明，一大类直角Artin群（特别是，那些与平面互补定义图）可以嵌入拟等距纯辫群和组的区域保持固定边界的磁盘的同构（相对于$L^2$ -范数度量），这延伸了Benaim和Gambaudo的结果谁给拟等距嵌入的$F\_n$和$\Z^n$的所有$n>0$。因此，我们也能够嵌入各种Gromov双曲群拟等距纯辫群和在的圆同态群的磁盘。例子包括双曲曲面群，这些曲面群沿循环子群的一些HNN-扩张和某个闭双曲3-流形的基本群。
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。
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 $VB_3'/VB_3'' =\mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus\mathbb{Z}_3 \oplus \mathbb{Z}^{\infty}$, $VB_4' / VB_4'' = \mathbb{Z}_3 \oplus \mathbb{Z}_3 \oplus \mathbb{Z}_3$, $WB…
Congruence Subgroups of the Virtual Braid Group
通过扩展整系数Burau表示，将辫群的同余子群概念推广到虚拟辫群，证明其二级同余子群为纯虚拟辫群，恢复Arnol'd结果的虚拟类比。
The classification and the conjugacy classes of the finite subgroups of the sphere braid groups
设n为\geq 3。对球面辫群B_n（S^2）的子群所实现的有限群进行了分类.这些群必须是上同调周期2或4的群。本文证明了B_n（S^2）的极大有限子群是：Z_{2(n-1)} ; 4 n阶和4（n-2）阶双循环群;二元四面体群T_1;二元八面体群O_1;二元二十面体群I。给出了这类有限子群在B_n（S^2）中的几何构造和一些显式代数构造，并确定了这类有限子群的共轭类的个数.我们还重新证明了Murasugi对B_n（S^2）的扭元的分类，并解释了B_n（S^2）的有限子群与这一分类的关系，以及与B_n（S^2）的下中心列和导出列的关系.
Algebraic structures among virtual singular braids
该研究证明n股虚拟奇异辫子幺半群可嵌入群VSGₙ，给出其半直积结构与VSPGₙ的表示及子群分解，为虚拟奇异辫子的代数结构分析提供基础。
Powers of half-twists and congruence subgroups of braid groups
本文研究辫群同余子群$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。
The classification of the virtually cyclic subgroups of the sphere braid groups
研究了2-球面S^2的n串辫群B_n（S^2）的虚循环子群的同构类的确定问题。如果n是奇数，或者n是偶数并且足够大，我们就得到了完全分类。对于小的偶数值的n，分类是完整的，直到一个明确的有限数量的开放情况。为了证明我们的主要定理，我们得到了一些其他的独立结果，特别是B_n（S^2）的有限循环子群和双循环子群的中心化子和正规化子的特征，关于有穷级元的共轭幂的结果，作为B_n（S^2）的子群出现的合并积的同构类的分析，以及当n大于或等于3时，S^2的n阶位形空间的万有覆叠空间具有S^3的同伦型的另一种证明.
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。
Abelianization of Subgroups of Reflection Group and their Braid Group; an Application to Cohomology
本文的最终结果给出了扩张$$\xymatrix{1\ar[r] & P/[P,P] \ar^{j}[r] & B/[P,P] \ar^-{p}[r] & W \ar[r] & 1}$$作为上同调群$H^2(W,P/[P,P])$的元素的阶（其中$B$和$P$代表与复反射群$W$相关联的辫群和纯辫群）。为了得到这个结果，我们描述了超平面$H$的稳定子$N_H$的阿贝尔化。与Coxeter群的情况相反，$N_H$一般不是复反射群$W$的反射子群。因此，第一步是完善斯坦利-施普林格关于反射群阿贝尔化的定理。第二步是描述$W$的辫群$B$的各种类型的大子群的阿贝尔化。更准确地说，我们只需要$p$从$N_H$的逆像得到一个群同态，其值在$\QQ$中（其中$p : B \ra W$是规范态射），但稍微增强就可以完整地描述$p^{-1}(W')$的阿贝尔化，其中$W'$是$W$的反射子群或超平面的稳定子。我们还建议在$W$中对反射的中心化器的每个元素进行提升构造。
Universal virtual braid groups
本文引入了通用虚拟编织群UV_n(c)，提供了一种统一的代数框架，用于虚拟编织结构，证明其具有残余有限性、线性、解字和共轭问题等性质，并分类了子群分离性和Howson性质。
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。
Let n be greater than or equal to 3. We study the quotient group B\_n/[P n,P\_n] of the Artin braid group B\_n by the commutator subgroup of its pure Artin braid group P\_n. We show that B\_n/[P n,P\_n] is a crystallographic group, and in the case n=3, we analyse explicitly some of its subgroups. We also prove that B\_n/[P n,P\_n] possesses torsion, and we show that there is a one-to-one correspondence between the conjugacy classes of the finite-order elements of B\_n/[P n,P\_n] with the conjugacy classes of the elements of odd order of the symmetric group S\_n, and that the isomorphism class …
Quotients of the braid group that are extensions of the symmetric group
本文研究辫群中使商为对称群阿贝尔扩张的正规子群，证明其恰有8个可公度类，定义Specht子群并刻画其模与上同调性质，确定扩张是否分裂。
The co-Hopfian property of surface braid groups
设g和n是至少为2的整数，G是亏格为g的闭可定向曲面上的n股纯辫群.我们刻画了从G的有限指数子群到G的任何单同态。因此，我们证明了G的任何有限指数子群都是余霍普夫子群。
Vassiliev invariants and knots modulo pure braid subgroups
我们证明了两个结有匹配的Vassiliev不变量的顺序小于n当且仅当它们是等价模的第n组的下中心序列的一些纯辫子群，从而表征Vassiliev的结不变量的结构方面的辫群。我们还证明了纯辫群的纽结模第n个导子群以及一般的纽结模辫子群的一些结果。
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。
Brunnian Braids on Surfaces
我们确定了一般曲面$M$上的Brunnian辫子的生成元集，其中的生成元为$M\not=S^2$或$\RP^2$。对于$M=S^2$或$\RP^2$的情形，我们的生成集与2-球面的同伦群一起给出了$M$上Brunnian辫子的生成元集.
By evaluating the Burau representation at t=-1, we obtain a symplectic representation of the braid group. We define the congruence subgroups of the braid group to be the preimages of the principal congruence subgroups of the symplectic group. Our main result is that the level four congruence subgroup of the braid group is equal to the group generated by squares of Dehn twists and is also equal to the group generated by squares of pure braids. We also compute the image of the point pushing subgroup under the symplectic representation.
Classifying spaces for families of virtually abelian subgroups of surface braid groups
本文研究了表面纯辫群虚阿贝尔子群族的分类空间的最小维数，证明其等于群的虚上同调维数加n，并应用于计算可算子群族的分类空间最小维数。
We study the centralizer of a braid from the point of view of Garside theory, showing that generically a minimal set of generators can be computed very efficiently, as the ultra summit set of a generic braid has a very particular structure. We present an algorithm to compute the centralizer of a braid whose generic-case complexity is quadratic on the length of the input, and which outputs a minimal set of generators in the generic case.
Braid pictures for Artin groups
我们定义了二维orbifold的辫群，并介绍了绘制辫图的惯例。我们利用它们实现了与球面Coxeter图A_n，B_n = C_n和D_n以及仿射图波浪线{A}_n，波浪线{B}_n，波浪线{C}_n和波浪线{D}_n相联系的Artin群作为各种简单轨道折叠的辫群的子群。D_n、波浪线{B}_n、波浪线{C}_n和波浪线{D}_n是新的情况。在每种情况下，阿廷群都是一个具有交换商的正规子群;在除代字号{A}_n之外的所有情况下，商都是有限的。我们通过在D_n型Artin群中进行一个非平凡的计算来说明我们的辫子演算的价值。
About presentations of braid groups and their generalizations
本文综述了辫子理论中一些较新的概念和结果，它们推广了辫子理论中的经典概念和结果。在这些概念中有各种部分辫子的逆幺半群。我们还观察到不同的介绍从标准的Artin介绍概括的辫子。也就是说，我们认为介绍了少量的发电机，Sergiescu图介绍和Birman-Ko-Lee介绍。V. V. Chaynikov关于奇异辫幺半群在Birman-Ko-Lee生成元中的字和共轭问题的工作也被描述。
Pure braids, a new subgroup of the mapping class group and finite type invariants
在研究曲面的映射类群M与同调三维球面的有限型不变量理论之间的关系时，映射类群的三个子群起着重要的作用。它们是Torelli群，Johnson子群K和一个新的子群L，其中包含K，定义为选择的拉格朗日子群的同源性的表面。在这项工作中，我们确定商L/K，在约翰逊和森田给出的M/K的精确描述。利用纯辫群在L中的一些自然嵌入和有限型不变量理论，我们还研究了L和K的下中心序列。
A toric braid is a braid whose closure is a torus link in $\mathbb{R}^3$. Manturov generalized toric braids that is called quasitoric braids and showed that the subset of quasitoric braids in the classical braid group is a subgroup of the braid group. We call this subgroup the quasitoric braid group. In this paper, we give two minimal generating sets for the quasitoric braid group and determine its abelianization. The minimalities of these two generating sets are obtained from a lower bound by the number of generators for the abelianization.
Let $SB_n$ be the singular braid group generated by braid generators $σ_i$ and singular braid generators $τ_i$, $1 \leq i \leq n-1$. Let $ST_n$ denote the group that is the kernel of the homomorphism that maps, for each $i$, $σ_i$ to the cyclic permutation $(i, i+1)$ and $τ_i$ to $1$. In this paper we investigate the group $ST_3$. We obtain a presentation for $ST_3$. We prove that $ST_3$ is isomorphic to the singular pure braid group $SP_3$ on $3$ strands. We also prove that the group $ST_3$ is semi-direct product of a subgroup $H$ and an infinite cyclic group, where the subgroup $H$ is an HNN…
In this article, we give a necessary and sufficient condition for embedding a finite index subgroup of Artin's braid group into the mapping class group of a connected orientable surface.
In this paper, we investigate the structure of the automorphism groups of pure braid groups. We prove that, for $n>3$, $\Aut(P_n)$ is generated by the subgroup $\Aut_c(P_n)$ of central automorphisms of $P_n$, the subgroup $\Aut(B_n)$ of restrictions of automorphisms of $B_n$ on $P_n$ and one extra automorphism $w_n$. We also investigate the lifting and extension problem for automorphisms of some well-known exact sequences arising from braid groups, and prove that that answers are negative in most cases. Specifically, we prove that no non-trivial central automorphism of $P_n$ can be extended to…
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 form: loops corresponding to the hyperplanes which come nearest the basepoint. Our results support the conjecture that motivated this study, the "monstrous proposal", which posits a relationship between this braid group and the monster finite simple group.
The n-strand braid group can be defined as the fundamental group of the configuration space of n unlabeled points in a closed disk based at a configuration where all n points lie in the boundary of the disk. Using this definition, the subset of braids that have a representative where a specified subset of these points remain pointwise fixed forms a subgroup isomorphic to a braid group with fewer strands. In this article, we generalize this phenomenon by introducing the notion of boundary braids. A boundary braid is a braid that has a representative where some specified subset of the points rem…
p-adic framed braids
本文定义了$p$ -进框架辫群${\mathcal F}_{\infty,n}$，它是模框架辫子的逆极限，并给出了${\mathcal F}_{\infty, n}$的拓扑生成元.我们还给出了$p$ -adic框架编织的几何解释。然后我们构造了一个$p$ -adic Yokonuma-Hecke代数${\rm Y}_{\infty,n}(u)$作为一族经典Yokonuma-Hecke代数的逆极限.它们是模框架辫群在二次关系上的幂等。我们还给出了${\rm Y}_{\infty,n}(u)$的拓扑生成元。最后，我们在这个新代数上构造了一个线性迹，它支持马尔可夫性。
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.
Hierarchical geometry and right-angled Artin groups in graph braid groups
本文在图辫群的万有覆盖上构造显式因子系统，建立分层双曲群结构，给出自由阿贝尔子群最大秩的有限公式，并证明所有RAAG均可无扭曲嵌入，且对图2-辫群给出二分图限制与图论判据。
Probabilistic Solutions of Equations in the Braid Group
给定一个方程组在一个“随机”生成的子群的辫群，我们展示了如何找到一个小的有序列表中的元素的子组，其中包含一个解决方案的方程具有显着的概率。此外，该解决方案很有可能是列表中的第一个。这给出了一个概率解决方案：共轭问题，组成员问题，最短的代表性的一个元素，和其他组合群论问题的随机子群的辫群。我们使用基于内存的扩展标准的基于长度的方法，原则上可以适用于任何组承认一个有效的，合理的行为长度函数。
In previous work, joint with Bux, Fluch, Marschler and Witzel, we proved that the braided Thompson groups are of type $\textrm{F}_\infty$. The proof utilized certain contractible cube complexes, which in this paper we prove are CAT(0). We then use this fact to compute the geometric invariants $Σ^m(F_{\textrm{br}})$ of the pure braided Thompson group $F_{\textrm{br}}$. Only the first invariant $Σ^1(F_{\textrm{br}})$ was previously known. A consequence of our computation is that as soon as a subgroup of $F_{\textrm{br}}$ containing the commutator subgroup $[F_{\textrm{br}},F_{\textrm{br}}]$ is f…
The crossing matrix of a braid on $N$ strands is the $N\times N$ integer matrix with zero diagonal whose $i,j$ entry is the algebraic number (positive minus negative) of crossings by strand $i$ over strand $j$ . When restricted to the subgroup of pure braids, this defines a homomorphism onto the additive subgroup of $N\times N$ symmetric integer matrices with zero diagonal--in fact, it represents the abelianization of this subgroup. As a function on the whole $N$-braid group, it is a derivation defined by the action of the symmetric group on square matrices. The set of all crossing matrices ca…
Let $n, k \geq 3$. In this paper, we analyse the quotient group $B\_n/Γ\_k(P\_n)$ of the Artin braid group $B\_n$ by the subgroup $Γ\_k(P\_n)$ belonging to the lower central series of the Artin pure braid group $P\_n$. We prove that it is an almost-crystallographic group. We then focus more specifically on the case $k=3$. If $n \geq 5$, and if $τ\in N$ is such that $gcd(τ, 6) = 1$, we show that $B\_n/Γ\_3 (P\_n)$ possesses torsion $τ$ if and only if $S\_n$ does, and we prove that there is a one-to-one correspondence between the conjugacy classes of elements of order $τ$ in $B\_n/Γ\_3 (P\_n)$ w…
We study the problem of finding generators for the fundamental group G of a space of the following sort: one removes a family of complex hyperplanes from n dimensional complex vector space, or n dimensional complex hyperbolic space, or the Hermitian symmetric space for O(2,n), and then takes the quotient by a discrete group $PΓ$. The classical example is the braid group, but there are many similar "braid-like" groups that arise in topology and algebraic geometry. Our main result is that if $PΓ$ contains reflections in the hyperplanes nearest the basepoint, and these reflections satisfy a certa…
Ng and Schauenburg proved that the kernel of a $(2+1)$-dimensional topological quantum field theory representation of $\mathrm{SL}(2, \mathbb{Z})$ is a congruence subgroup. Motivated by their result, we explore when the kernel of an irreducible representation of the braid group $B_3$ with finite image enjoys a congruence subgroup property. In particular, we show that in dimensions two and three, when the projective order of the image of the braid generator $σ_1$ is between 2 and 5 the kernel projects onto a congruence subgroup of $\mathrm{PSL}(2,\mathbb{Z})$ and compute its level. However, we …
The proof of Birman's conjecture on singular braid monoids
设B_n是n个弦上的Artin辫群，其标准生成元为sigma_1，.，sigma_{n-1}，设SB_n是具有生成元σ_1 ^{+-1}，.的奇异辫幺半群，sigma_{n-1} ^{+-1}，tau_1，.，tau_{n-1}。去奇异化映射是乘法同态eta：SB_n --> Z[B_n]，定义为eta（sigma_i^{+-1}）=_i^{+-1}和eta（tau_i）= sigma_i - sigma_i^{-1}，其中1 <= i <= n-1。本文证明了Birman猜想，即去奇异化映射eta是单射的.
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