Lower central series for surface braid groups

The lower central and derived series of the braid groups of the projective plane

We determine the lower central and derived series of the n-string braid groups B_n(RP^2) of the real projective plane. We are motivated in part by the study of Fadell-Neuwirth short exact sequences, but the problem is interesting in its own right. For n=1,2, B_n(RP^2) is finite and its lower central and derived series are known. If n>2 (resp. n>4), we show that the lower central (resp. derived) series of B_n(RP^2) is constant from the commutator subgroup onwards, and we exhibit a presentation of the commutator subgroup. In the exceptional cases n=3,4, we determine explicitly the complete deriv…

当下中心列停止时：辫群及其相关群的全面研究

When the lower central series stops: a comprehensive study for braid groups and their relatives

本文研究群的下中心列何时停止的问题，引入多种工具并应用于辫群及其相关群，除射影平面上的一族分割辫群外均给出答案，部分情形完整计算了下中心列。

The lower central and derived series of the braid groups of the sphere and the punctured sphere

Our aim is to determine the lower central series (LCS) and derived series (DS) for the braid groups of the sphere and of the finitely-punctured sphere. We show that for all n (resp. all n\geq 5), the LCS (resp. DS) of the n-string braid group B\_n(S^2) is constant from the commutator subgroup onwards, and that Γ\_2(B\_4(S^2)) is a semi-direct product of the quaternion group by a free group of rank 2. For n=4, we determine the DS of B\_4(S^2), as well as its quotients. For n \geq 1, the class of m-string braid groups B\_m(S^2) \ {x\_1,...,x\_n} of the n-punctured sphere includes the Artin braid…

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。

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…

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。

Automorphisms of pure braid Groups

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…

The pure braid groups and their relatives

In this mostly survey paper, we investigate the resonance varieties, the lower central series ranks, and the Chen ranks, as well as the residual and formality properties of several families of braid-like groups: the pure braid groups $P_n$, the welded pure braid groups $wP_n$, the virtual pure braid groups $vP_n$, as well as their `upper' variants, $wP_n^+$ and $vP_n^+$. We also discuss several natural homomorphisms between these groups, and various ways to distinguish among the pure braid groups and their relatives.

Tangling and Braiding the Chessboard Complex

We describe a series of complexes that relate to the braid groups as the matching complexes relate to the symmetric groups. A modified construction applies as well to other complexes based on edge sets in graphs. We show that our constructions will yield Cohen-Macauley complexes provided the underlying complexes are Cohen-Macauley. Finally, we discuss a related series of complexes to provide some positive evidence that the braided Houghton groups, introduced by F. Degenhardt, are a series of groups with linearly increasing finiteness length as are the (unbraided) Houghton groups.

Conjugacy Classes of 3-Braid Group B_3

In this article we describe the summit sets in B_3, the smallest element in a summit set and we compute the Hilbert series corresponding to conjugacy classes.The results will be related to Birman-Menesco classification of knots with braid index three or less than three.

On p-almost direct products and residual properties of pure braid groups of nonorientable surfaces

We prove that the n th pure braid group of a nonorientable surface (closed or with boundary, but different from RP2) is residually 2-finite. Consequently, this group is residually nilpotent. The key ingredient in the closed case is the notion of p-almost direct product, which is a generalization of the notion of almost direct product. We prove therefore also some results on lower central series and augmentation ideals of p-almost direct products.

The 4-string Braid group $B_4$ has property RD and exponential mesoscopic rank

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 (of dimension 3). Namely, we show that both $B_4$ and its central quotient have exponential mesoscopic rank, i.e., that they contain exponentially many large flat balls which are not included in flats.

Pure virtual braids, resonance, and formality

We investigate the resonance varieties, lower central series ranks, and Chen ranks of the pure virtual braid groups and their upper-triangular subgroups. As an application, we give a complete answer to the 1-formality question for this class of groups. In the process, we explore various connections between the Alexander-type invariants of a finitely generated group and several of the graded Lie algebras associated to it, and discuss possible extensions of the resonance-Chen ranks formula in this context.

Basic results on braid groups

These are Lecture Notes of a course given by the author at the French-Spanish School "Tresses in Pau", held in Pau (France) in October 2009. It is basically an introduction to distinct approaches and techniques that can be used to show results in braid groups. Using these techniques we provide several proofs of well known results in braid groups, namely the correctness of Artin's presentation, that the braid group is torsion free, or that its center is generated by the full twist. We also recall some solutions of the word and conjugacy problems, and that roots of a braid are always conjugate. …

On injective homomorphisms for pure braid groups, and associated Lie algebras

The question of whether a representation of Artin's pure braid group is faithful is translated to certain properties of the Lie algebra arising from the descending central series of the pure braid group, and thus the Vassiliev invariants of pure braids via work of T. Kohno \cite{kohno1,kohno2}. The main result is a Lie algebraic condition which guarantees that a homomorphism out of the classical pure braid group is faithful. However, it is unclear whether the methods here can be applied to any open cases such as the Gassner representation.

Linear-central filtrations and the image of the Burau representation

The Burau representation is a fundamental bridge between the braid group and diverse other topics in mathematics. A 1974 question of Birman asks for a description of the image; in this paper we give a "strong approximation" to the answer. Since a 1984 paper of Squier it has been known that the Burau representation preserves a certain Hermitian form. We show that the Burau image is dense in this unitary group relative to a topology induced by a naturally-occurring filtration. We expect that the methods of the paper should extend to many other representations of the braid group and perhaps ultim…

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…

A kernel of a braid group representation yields a knot with trivial knot polynomials

We show that a non-trivial, non-central normal subgroup of the braid groups contains a braid whose closure is a hyperbolic knot with arbitrary large genus. This shows that non-faithfulness of a quantum representation implies that the corresponding quantum invariant fails to detect the unknot. The proof utilizes the Dehornoy ordering of the braid groups.

Goussarov-Polyak-Viro's $n$-equivalence and the pure virtual braid group

In the context of finite type invariants, Stanford introduced a family of equivalence relations on knots defined by the lower central series of the pure braid groups and characterized the finite type invariants in terms of the structure of the braid groups. It is known that this equivalence and Ohyama's equivalence defined by a local move are equivalent. On the other hand, in the virtual knot theory, the concept of Ohyama's equivalence was extended by Goussarov-Polyak-Viro, which called an $n$-equivalence. In this paper we extend Stanford's equivalence to virtual knots and virtual string links…

Conjugacy in Garside Groups III: Periodic braids

An element in Artin's braid group B_n is said to be periodic if some power of it lies in the center of B_n. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B_n are exponential in the braid index n for the special case of periodic braids. We overcome this difficulty by putting to work several known isomorphisms between Garside structures in the braid group B_n and other Garside groups. This allows us to obtain a polynomial solution to the original problem in the spirit of the previously known algorithms. This paper is the third in a series…

Vassiliev invariants and knots modulo pure braid subgroups

We show that two knots have matching Vassiliev invariants of order less than n if and only if they are equivalent modulo the nth group of the lower central series of some pure braid group, thus characterizing Vassiliev's knot invariants in terms of the structure of the braid groups. We also prove some results about knots modulo the nth derived subgroups of the pure braid groups, and about knots modulo braid subgroups in general.

The Fundamental Group of $SO(n)$ Via Quotients of Braid Groups

We describe an algebraic proof of the well-known topological fact that $π_1(SO(n)) \cong Z/2Z$. The fundamental group of $SO(n)$ appears in our approach as the center of a certain finite group defined by generators and relations. The latter is a factor group of the braid group $B_n$, obtained by imposing one additional relation and turns out to be a nontrivial central extension by $Z/2Z$ of the corresponding group of rotational symmetries of the hyperoctahedron in dimension $n$.

Braids: A Survey

This article is about Artin's braid group and its role in knot theory. We set ourselves two goals: (i) to provide enough of the essential background so that our review would be accessible to graduate students, and (ii) to focus on those parts of the subject in which major progress was made, or interesting new proofs of known results were discovered, during the past 20 years. A central theme that we try to develop is to show ways in which structure first discovered in the braid groups generalizes to structure in Garside groups, Artin groups and surface mapping class groups. However, the literat…

Hurwitz Equivalence of Braid Group Factorizations Consisting of a Semi-Frame

In this paper we prove certain Hurwitz equivalence properties in the braid group. Our main result is that every two factorizations of $Δ_n ^2$ where the elements of the factorization are semi-frame are Hurwitz equivalent. The results of this paper are generalization of the results in \cite{B4}. We use a new presentation of the braid group, called the Birman-Ko-Lee presentation, to define the semi-frame structure. The main result of this paper can be applied to compute the BMT invariant of surfaces. The BMT is the class of Hurwitz equivalent factorizations of the central element of the braid gr…

A remarkable family of left ordered groups: central extensions of Hecke groups

We provide an infinite family of left-ordered groups, all of which have a positive cone that is finitely generated as a semigroup. This family includes the Klein bottle group and the braid group B_3.

Local braid monodromies and local fundamental groups of tangented conic-line arrangements

This paper is the first part of a series of three papers about the fundamental groups of conic-line arrangements consist of two tangented conics and up to two additional tangented lines. In this part, we compute the local braid monodromies and the local fundamental groups of the singularities which appear in such arrangements.

The braid group $B_3$ in the framework of continued fractions

We use the classical interpretation of the braid group $B_3$ as a central extension of the modular group $\text{PSL}_2\left(\mathbb{Z}\right)$ to establish new and fundamental properties of $B_3$ using the theory of continued fractions. In particular, we give simple and natural linear time algorithms to solve the word and conjugacy problems in $B_3$. The algorithms introduced in this paper are easy to implement and are the most efficient algorithms in the literature to solve these problems in the braid group $B_3$.

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 …

Manifolds of Triangulations, braid groups of manifolds and the groups $Γ_{n}^{k}$

The spaces of triangulations of a given manifold have been widely studied. The celebrated theorem of Pachner~\cite{Pachner} says that any two triangulations of a given manifold can be connected by a sequence of bistellar moves, or Pachner moves, see also~\cite{GKZ,Nabutovsky}. In the present paper we consider groups which naturally appear when considering the set of triangulations with fixed number of simplices of maximal dimension. There are three ways of introducing this groups: the geometrical one, which depends on the metric, the topological one, and the combinatorial one. The second one c…

Almost-crystallographic groups as quotients of Artin braid groups

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…

The classification of the virtually cyclic subgroups of the sphere braid groups

We study the problem of determining the isomorphism classes of the virtually cyclic subgroups of the n-string braid groups B_n(S^2) of the 2-sphere S^2. If n is odd, or if n is even and sufficiently large, we obtain the complete classification. For small even values of n, the classification is complete up to an explicit finite number of open cases. In order to prove our main theorem, we obtain a number of other results of independent interest, notably the characterisation of the centralisers and normalisers of the finite cyclic and dicyclic subgroups of B_n(S^2), a result concerning conjugate …

不可定向曲面上辫群的几何表示

Geometric representations of the braid group on a nonorientable surface

本文分类了从 $n$ 股辫群到亏格为 $g$ 的不可定向曲面纯映射类群的同态，证明了在特定参数条件下此类同态必为循环映射或映射至特定几何元素。

Abelianization of Subgroups of Reflection Group and their Braid Group; an Application to Cohomology

The final result of this article gives the order of the extension $$\xymatrix{1\ar[r] & P/[P,P] \ar^{j}[r] & B/[P,P] \ar^-{p}[r] & W \ar[r] & 1}$$ as an element of the cohomology group $H^2(W,P/[P,P])$ (where $B$ and $P$ stands for the braid group and the pure braid group associated to the complex reflection group $W$). To obtain this result, we describe the abelianization of the stabilizer $N_H$ of a hyperplane $H$. Contrary to the case of Coxeter groups, $N_H$ is not in general a reflection subgroup of the complex reflection group $W$. So the first step is to refine Stanley-Springer's theore…

Pure braids, a new subgroup of the mapping class group and finite type invariants

In the study of the relation between the mapping class group M of a surface and the theory of finite-type invariants of homology 3-spheres, three subgroups of the mapping class group play a large role. They are the Torelli group, the Johnson subgroup K and a new subgroup L, which contains K, defined by a choice of a Lagrangian subgroup of the homology of the surface. In this work we determine the quotient L/K, in terms of the precise description of M/K given by Johnson and Morita. We also study the lower central series of L and K, using some natural imbeddings of the pure braid group in L and …

A theory of orbit braids

This paper upbuilds the theoretical framework of orbit braids in $M\times I$ by making use of the orbit configuration space $F_G(M,n)$, which enriches the theory of ordinary braids, where $M$ is a connected topological manifold of dimension at least 2 with an effective action of a finite group $G$ and the action of $G$ on $I$ is trivial. Main points of our work include as follows. We introduce the orbit braid group $\mathcal{B}_n^{orb}(M,G)$, and show that it is isomorphic to a group with an additional endowed operation (called the extended fundamental group of $F_G(M,n)$), formed by the homot…

lower central series 和 split extensions

Lower central series and split extensions

本文研究了群的N-级数及其关联的李代数，探讨了split extensions的lower central series及其扩展版本的性质，提出了P-LCS的通用构造，并应用于表面辫群的残余nilpotency研究。

球型 Artin-Tits 群中的典范约化系统

Canonical Reduction Systems in Artin-Tits groups of spherical type

本文在球型 Artin-Tits 群中引入元素的典范约化系统，推广了辫群中的概念，证明了其基本性质，给出了计算算法，并证明了中心化子序列的周期性。

Central extensions of the Ptolemy-Thompson group and quantized Teichmuller theory

The central extension of the Thompson group $T$ that arises in the quantized Teichmüller theory is 12 times the Euler class. This extension is obtained by taking a (partial) abelianization of the so-called braided Ptolemy-Thompson group introduced and studied in \cite{FK2}. We describe then the cyclic central extensions of $T$ by means of explicit presentations.

Topological Entropy of Surface Braids and Maximally Efficient Mixing

The deep connections between braids and dynamics by way of the Nielsen-Thurston classification theorem have led to a wide range of practical applications. Braids have been used to detect coherent structures and mixing regions in oceanic flows, drive the design of industrial mixing machines, contextualize the evolution of taffy pullers, and characterize the chaotic motion of topological defects in active nematics. Mixing plays a central role in each of these examples, and the braids naturally associated with each system come equipped with a useful measure of mixing efficiency, the topological e…

Divisor braids

We study a novel type of braid groups on a closed orientable surface $Σ$. These are fundamental groups of certain manifolds that are hybrids between symmetric products and configuration spaces of points on $Σ$; a class of examples arises naturally in gauge theory, as moduli spaces of vortices in toric fibre bundles over $Σ$. The elements of these braid groups, which we call divisor braids, have coloured strands that are allowed to intersect according to rules specified by a graph $Γ$. In situations where there is more than one strand of each colour, we show that the corresponding braid group a…

Families of superelliptic curves, complex braid groups and generalized Dehn twists

We consider the universal family $E_n^d$ of superelliptic curves: each curve $Σ_n^d$ in the family is a $d$-fold covering of the unit disk, totally ramified over a set $P$ of $n$ distinct points; $Σ_n^d\hookrightarrow E_n^d\to C_n$ is a fibre bundle, where $C_n$ is the configuration space of $n$ distinct points. We find that $E_n^d$ is the classifying space for the complex braid group of type $B(d,d,n)$ and we compute a big part of the integral homology of $E_n^d,$ including a complete calculation of the stable groups over finite fields by means of Poincarè series. The computation of the main …

Lawrence-Bigelow representations, bases and duality

We study homological representations of mapping class groups, including the braid groups. These arise from the twisted homology of certain configuration spaces, and come in many different flavours. Our goal is to give a unified general account of the fundamental relationships (non-degenerate pairings, embeddings, isomorphisms) between the many different flavours of homological representations. Our motivating examples are the Lawrence-Bigelow representations of the braid groups, which are of central importance in the study of the braid groups themselves, as well as their connections with quantu…

The dilogarithmic central extension of the Ptolemy-Thompson group via the Kashaev quantization

Quantization of universal Teichmüller space provides projective representations of the Ptolemy-Thompson group, which is isomorphic to the Thompson group $T$. This yields certain central extensions of $T$ by $\mathbb{Z}$, called dilogarithmic central extensions. We compute a presentation of the dilogarithmic central extension $\hat{T}^{Kash}$ of $T$ resulting from the Kashaev quantization, and show that it corresponds to $6$ times the Euler class in $H^2(T;\mathbb{Z})$. Meanwhile, the braided Ptolemy-Thompson groups $T^*$, $T^\sharp$ of Funar-Kapoudjian are extensions of $T$ by the infinite bra…

Large color $R$-matrix for knot complements and strange identities

The Gukov-Manolescu series, denoted by $F_K$, is a conjectural invariant of knot complements that, in a sense, analytically continues the colored Jones polynomials. In this paper we use the large color $R$-matrix to study $F_K$ for some simple links. Specifically, we give a definition of $F_K$ for positive braid knots, and compute $F_K$ for various knots and links. As a corollary, we present a class of `strange identities' for positive braid knots.

流环与量子群

Flow loops and quantum groups

本文通过研究结的量子群不变量与莫尔斯流的动力学，定义了一个双变量级数不变量，用于描述结补集中的莫尔斯流环，推测该动力学级数与结补集的BPS q级数一致，后者源于量子群的Verma模，编码所有着色Jones多项式。