David Bessis, Jean Michel
We give presentations for the braid groups associated with the complex reflection groups $G_{24}$ and $G_{27}$. For the cases of $G_{29}$, $G_{31}$, $G_{33}$ and $G_{34}$, we give (strongly supported) conjectures. These presentations were obtained with VKCURVE, a GAP package implementing Van Kampen's method.
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In his initial paper on braids E.Artin gave a presentation with two generators for an arbitrary braid group. We give analogues of this Artin's presentation for various generalizations of braids.
Presentations are computed for a braided version BV of Thompson's group V and for V itself showing that there is an Artin group/Coxeter group relation between them. The presentation for V is obtained from that for BV by declaring all that all generators are involutions.
Birman, Ko and Lee have introduced a new monoid ${\cal B}^{*}_{n}$--with an explicit presentation--whose group of fractions is the $n$-strand braid group ${\cal B}_{n}$. Building on a new approach by Digne, Michel and himself, Bessis has defined a {\it dual} braid monoid for every finite Coxeter type Artin-Tits group extending the type A case. Here, we give an explicit presentation for this dual braid monoid in the case of types B and D, and we study the combinatorics of the underlying Garside structures.
Let G be a graph. The (unlabeled) configuration space of n points on G is the space of all n-element subsets of G. The fundamental group of such a configuration space is called a graph braid group. We use a version of discrete Morse theory to compute presentations of all graph braid groups, for all finite connected graphs G and all natural numbers n.
We describe a new presentation for the complex reflection groups of type $(e,e,r)$ and their braid groups. A diagram for this presentation is proposed. The presentation is a monoid presentation which is shown to give rise to a Garside structure. A detailed study of the combinatorics of this structure leads us to describe it as post-classical.
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.
We provide new group presentations for surface braid groups which are positive. We study some properties of such presentations and we solve the conjugacy problem in a particular case.
We obtain new presentations for the imprimitive complex reflection groups of type $(de,e,r)$ and their braid groups $B(de,e,r)$ for $d,r \ge 2$. Diagrams for these presentations are proposed. The presentations have much in common with Coxeter presentations of real reflection groups. They are positive and homogeneous, and give rise to quasi-Garside structures. Diagram automorphisms correspond to group automorphisms. The new presentation shows how the braid group $B(de,e,r)$ is a semidirect product of the braid group of affine type $\widetilde A_{r-1}$ and an infinite cyclic group. Elements of $…
In this paper we give new presentations of the braid groups and the pure braid groups of a closed surface. We also give an algorithm to solve the word problem in these groups, using the given presentations.
The aim of this article is to prove that the kernel of the map from the pure braid group $PB_{n},n\ge 4$ to the group $G_{n}^{3}$ consists of full twist braids and their exponents. The proof consists of two parts. The first part which deals with $n=4$ relies on the crucial tool in this construction having its own interest is the {\em realisability projection} saying that if two {\em realisable} $G_{4}^{3}$-elements are equivalent then they are equivalent by a sequence of realisable ones. The second part (an easy one) uses induction on $n$.
In the paper we give a survey of rather new notions and results which generalize classical ones in the theory of braids. Among such notions are various inverse monoids of partial braids. We also observe presentations different from standard Artin presentation for generalizations of braids. Namely, we consider presentations with small number of generators, Sergiescu graph-presentations and Birman-Ko-Lee presentation. The work of V.V.Chaynikov on the word and conjugacy problems for the singular braid monoid in Birman-Ko-Lee generators is described as well.
We give presentations, in terms of generators and relations, for the monoids of singular braids on closed surfaces. The proof of the validity of these presentations can also be applied to verify, in a new way, the presentations given by Birman for the monoids of Singular Artin braids.
We give several new positive finite presentations for the pure braid group that are easy to remember and simple in form. All of our presentations involve a metric on the punctured disc so that the punctures are arranged "convexly", which is why we describe them as geometric presentaitons. Motivated by a presentation for the full braid group that we call the "rotation presentation", we introduce presentations for the pure braid group that we call the "twist presentation" and the "swing presentation". From the point of view of mapping class groups, the swing presentation can be interpreted as st…
A large class of positive finite presentations of the braid groups is found and studied. It is shown that no presentations but known exceptions in this class have the property that equivalent braid words are also equivalent under positive relations.
The braid group $B_n$ maps homomorphically into the Temperley-Lieb algebra $\TL_n$. It was shown by Zinno that the homomorphic images of simple elements arising from the dual presentation of the braid group $B_n$ form a basis for the vector space underlying the Temperley-Lieb algebra $\TL_n$. In this paper, we establish that there is a dual presentation of Temperley-Lieb algebras that corresponds to the dual presentation of braid groups, and then give a simple geometric proof for Zinno's theorem, using the interpretation of simple elements as non-crossing partitions.
Quasipositive Braids, Presentations, and Garside Structures
本文给出辫子群的一种特定表示，证明其正幺半群不与拟正辫子幺半群同构，且拟正辫子幺半群无真Garside族，解决了Calegari的一个问题。
We describe presentations of braid groups of type $A$ arising from coloured quivers of mutation type $A$. We show that these can be interpreted geometrically as generalised triangulations of regular polygons.
We show that the Steinberg group $\text{St}(C_2,{\mathbb Z})$ associated with the Lie type $C_2$ and with integer coefficients can be realized as a quotient of the braid group $B_6$ by one relation. As an application we give a new braid-like presentation of the symplectic modular group $\text{Sp}_4({\mathbb Z})$.
If G is a finite graph and n is a natural number, then the n-strand braid group of G is the fundamental group of the configuration space of n points on G. This article gives a complete computation of the integral cohomology rings of the n-strand braid groups in case G is a tree. Some of the argument builds on earlier joint work with Lucas Sabalka.
We give a new presentation of the braid group $B$ of the complex reflection group $G(e,e,r)$ which is positive and homogeneous, and for which the generators map to reflections in the corresponding complex reflection group. We show that this presentation gives rise to a Garside structure for $B$ with Garside element a kind of generalised Coxeter element, and hence obtain solutions to the word and conjugacy problems for $B$.
A presentation for the planar pure braid group
本文构造了 $n$ 股平面纯辫群 $\overline{P}_n$ 的极小展示，其生成元与 Y.Baryshnikov 发现的上同调环生成元对偶，关系位于换位子群中，并给出了纯仙人掌群的显式非极小展示。
Hughes defined a class of groups that act as local similarities on compact ultrametric spaces. Guba and Sapir had previously defined braided diagram groups over semigroup presentations. The two classes of groups share some common characteristics: both act properly by isometries on CAT(0) cubical complexes, and certain groups in both classes have type F-infinity, for instance. Here we clarify the relationship between these families of groups: the braided diagram groups over tree-like semigroup presentations are precisely the groups that act on compact ultrametric spaces via small similarity str…
Pure Braid Group Presentations via Longest Elements
本文给出了经典纯辫群的一个新的简化展示，生成元由连通子图上最长元素的平方给出，关系仅为交换子或长度为5的回文盒关系，并证明了类似集合并非对所有Coxeter配置都生成，从而回答了Donovan和Wemyss的问题。
We describe presentations of braid groups of type ADE and show how these presentations are compatible with mutation of quivers, building on work of Barot and Marsh for Coxeter groups. In types A and D these presentations can be understood geometrically using triangulated surfaces. We then give a categorical interpretation of the presentations, with the new generators acting as spherical twists at simple modules on derived categories of Ginzburg dg-algebras of quivers with potential.
A lexicographic section of the braid arrangement and the modified Artin presentation
本文研究了辫子排列的词典部分，并通过辫子单胞计算其补集的基本群，证明其与修改的Artin展示一致，并扩展到更高类比的Manin-Schechtman排列。
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.
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…
We generalize presentations of the fundamental group of discriminant complements and arrive at a class of presentations associated naturally with words in the free monoid of the alphabet $σ_1,\dots,σ_{n-1}$. Our study addresses invariance properties of these presentations and the presented groups under various operations on the words. In particular we prove that the group does only depend on the corresponding element in the positive braid monoid, and under mild hypotheses only on its conjugacy class in the braid group.
This paper dates back to 1999 but was never published. The major part of it was included in the joint paper [Digne-Gomi, Presentation of pure braid groups, J. Knot Theory and its Ramifications 10 (2001) 609--623]. Sections 2 and 6 were not included there. They are independent from the rest of the paper. In section 2 we define a morphism from a braid group B associated to a Coxeter Group W to the semi-direct product of ZT (where T is the set of reflections) by W. This morphism lifts the map N from W to (Z/2Z)T defined by Dyer in [Reflection subgroups of Coxeter systems, J. Algebra 135 (1990) 57…
In this paper we give a description of the generators of the prime level congruence subgroups of braid groups. Also, we give a new presentation of the symplectic group over a finite field, and we calculate symmetric quotients of the prime level congruence subgroups of braid groups. Finally, we find a finite generating set for the level-3 congruence subgroup of the braid group on 3 strands.
The submonoid of the $3$-strand braid group $\mathcal{B}_3$ generated by $σ_1$ and $σ_1 σ_2$ is known to yield an exotic Garside structure on $\mathcal{B}_3$. We introduce and study an infinite family $(M_n)_{n\geq 1}$ of Garside monoids generalizing this exotic Garside structure, i.e., such that $M_2$ is isomorphic to the above monoid. The corresponding Garside group $G(M_n)$ is isomorphic to the $(n,n+1)$-torus knot group-which is isomorphic to $\mathcal{B}_3$ for $n=2$ and to the braid group of the exceptional complex reflection group $G_{12}$ for $n=3$. This yields a new Garside structure …
We consider the Lie algebra associated with the descending central series filtration of the pure braid group of a closed surface of arbitrary genus. R. Bezrukavnikov gave a presentation of this Lie algebra over the rational numbers. We show that his presentation remains true for this Lie algebra itself, i.e. over integers.
Presentations of the braid group of the complex reflection group $G(d,d,n)$
本文证明了复反射群 $G(d,d,n)$ 的辫群是某轨道折叠商空间辫群的指数 $d$ 子群，并给出了与标记三角剖分相容的表示。
Generalization of the Dehornoy-Lafont order complex to categories. Application to exceptional braid groups
本文将Dehornoy-Lafont序复形推广到Garside范畴并提出降时计算技术，用以补全例外复辫子群的同调结果，重点研究$B(G_{31})$。
We describe pure braided versions of Thompson's group F. These groups, $BF$ and $\hat{BF}$, are subgroups of the braided versions of Thompson's group V, introduced by Brin and Dehornoy. Unlike V, elements of F are order-preserving self-maps of the interval and we use pure braids together with elements of F thus preserving order. We define these groups and give normal forms for elements and describe infinite and finite presentations of these groups.
In this paper we study abelian and metabelian quotients of braid groups on oriented surfaces with boundary components. We provide group presentations and we prove rigidity results for these quotients arising from exact sequences related to (generalised) Fadell-Neuwirth fibrations.
Consider the ring R:=\Q[τ,τ^{-1}] of Laurent polynomials in the variable τ. The Artin's Pure Braid Groups (or Generalized Pure Braid Groups) act over R, where the action of every standard generator is the multiplication by τ. In this paper we consider the cohomology of these groups with coefficients in the module R (it is well known that such cohomology is strictly related to the untwisted integral cohomology of the Milnor fibration naturally associated to the reflection arrangement). We compute this cohomology for the cases I_2(m), H_3, H_4, F_4 and A_n with 1<n<7.
In this article we prove theorem on Lifting for the set of virtual pure braid groups. This theorem says that if we know presentation of virtual pure braid group $VP_4$, then we can find presentation of $VP_n$ for arbitrary $n > 4$. Using this theorem we find the set of generators and defining relations for simplicial group $T_*$ which was defined in the previuos article of the authors. We find a decomposition of the Artin pure braid group $P_n$ in semi-direct product of free groups in the cabled generators.
Inverse braid monoid describes a structure on braids where the number of strings is not fixed. So, some strings of initial $n$ may be deleted. In the paper we show that many properties and objects based on braid groups may be extended to the inverse braid monoids. Namely we prove an inclusion into a monoid of partial monomorphisms of a free group. This gives a solution of the word problem. Another solution is obtained by an approach similar to that of Garside. We give also the analogues of Artin presentation with two generators and Sergiescu graph-presentations.
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.
The word problem of a group is a very important question. The word problem in the braid group is of particular interest for topologists, algebraists and geometers. In previouse article we have looked at the braid group from a topological point of view, and thus using a new computerized representation of some elements of the fundamental group we gave a solution for its word problem. In this paper we will give an algorithm that will make it possible to transform the new presentation into a syntactic presentation. This will make it possible to computerize the group operation to sets of elements o…
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.
We give formulae for the first homology of the $n$-braid group and the pure 2-braid group over a finite graph in terms of graph theoretic invariants. As immediate consequences, a graph is planar if and only if the first homology of the $n$-braid group over the graph is torsion-free and the conjectures about the first homology of the pure 2-braid groups over graphs in \cite{FH} can be verified. We discover more characteristics of graph braid groups: the $n$-braid group over a planar graph and the pure 2-braid group over any graph have a presentation whose relators are words of commutators, and …
In this article, we investigate various properties of the pure virtual braid group PV_3. From its canonical presentation, we obtain a free product decomposition of PV_3. As a consequence, we show that PV_3 is residually torsion free nilpotent, which implies that the set of finite type invariants in the sense of Goussarov-Polyak-Viro is complete for virtual pure braids with three strands. Moreover we prove that the presentation of PV_3 is aspherical. Finally we determine the cohomology ring and the associated graded Lie algebra of PV_3.
We find finite presentations for the automorphism group of the Artin pure braid group and the automorphism group of the pure braid group associated to the full monomial group.
In the paper we give a survey on braid groups and subjects connected with them. We start with the initial definition, then we give several interpretations as well as several presentations of these groups. Burau presentation for the pure braid group and the Markov normal form are given next. Garside normal form and his solution of the conjugacy problem are presented as well as more recent results on the ordering and on the linearity of braid groups. Next topics are the generalizations of braids, their homological properties and connections with the other mathematical fields, like knot theory (v…
Permutahedron Triangulations via Total Linear Stability and the Dual Braid Group
本文为每个有限考克斯特群及其标准考克斯特元素构造了置换多面体的三角剖分，证明特定实现下该剖分由全线性稳定性高度函数诱导的正则三角剖分，并揭示其与Bruhat偏序、非交叉划分格及Cambrian同余的组合联系，进而为统一证明纯辫子群的简单类型一致表示提供机制。
It has been conjectured that in a braid group, or more generally in a Garside group, applying any sequence of monotone equivalences and word reversings can increase the length of a word by at most a linear factor depending on the group presentation only. We give a counter-example to this conjecture, but, on the other hand, we establish length upper bounds for the case when only right reversing is involved. We also state a new conjecture which would, like the above one, imply that the space complexity of the handle reduction algorithm is linear.
Springer categories for regular centralizers in well-generated complex braid groups
本文研究了复杂辫群中正则中心化子的弹簧类别，通过Garside结构证明了关于中心的定理，并提供了Hurwitz-like的呈现方法。
We complete the computation of the integral homology of the generalized braid group $B$ associated to an arbitrary irreducible complex reflection group $W$ of exceptional type. In order to do this we explicitely computed the recursively-defined differential of a resolution of $\mathbf{Z}$ as a $\mathbf{Z} B$-module, using parallel computing. We also deduce from this general computation the rational homology of the Milnor fiber of the singularity attached to most of these reflection groups.
We define pseudo-Garside groups and prove a theorem about them parallel to Garside's result on the word problem for the usual braid groups. The main novelty is that the set of simple elements can be infinite. We introduce a group B=B(Z^n) which we call the braid group of Z^n, and which bears some vague resemblance to mapping class groups. It is to GL(n,Z) what the braid group is to the symmetric group S_n. We prove that B is a pseudo-Garside group. We give a small presentation for B(Z^n) assuming one for B(Z^3) is given.
Consider the unit ball, $B = D \times [0,1]$, containing $n$ unknotted arcs $a_1, a_2, ..., a_n$ such that the boundary of each $a_i$ lies in $D \times \{0\}$. The Hilden (or Wicket) group is the mapping class group of $B$ fixing the arcs $a_1 \cup a_2 \cup ... \cup a_n$ setwise and fixing $D \times \{1\}$ pointwise. This group can be considered as a subgroup of the braid group. The pure Hilden group is defined to be the intersection of the Hilden group and the pure braid group. In a previous paper we computed a presentaion for the Hilden group using an action of the group on a cellular comple…
We consider the relative group presentation $\mathcal{P} = \langle G, \mathbf{x} | \mathbf{r} \rangle$ where $\mathbf{x} = \{ x \}$ and $\mathbf{r} = \{ xg_1 xg_2 xg_3 x^{-1} g_4 \}$. We show modulo a small number of exceptional cases exactly when $\mathcal{P}$ is aspherical. If $H = \langle g_1^{-1} g_2, g_1^{-1} g_3 g_1 , g_4 \rangle \leq G$ then the exceptional cases occur when $H$ is isomorphic to one of $C_5,C_6,C_8$ or $C_2 \times C_4$.
In this paper we show that the singular braid monoid of an orientable surface can be embedded in a group. The proof is purely topological, making no use of the monoid presentation.
The growth function is the generating function for sizes of spheres around the identity in Cayley graphs of groups. We present a novel method to calculate growth functions for automatic groups with normal form recognizing automata that recognize a single normal form for each group element, and are at most context free in complexity: context free grammars can be translated into algebraic systems of equations, whose solutions represent generating functions of their corresponding non-terminal symbols. This approach allows us to seamlessly introduce weightings on the growth function: assign differ…
This article is dedicate to cabling on virtual braids. This construction gives a new generating set for the virtual pure braid group $VP_n$. Consequently we describe $VP_4$ as HNN-extension. As an application to classical braids, we find a new presentation of the Artin pure braid group $P_4$ in terms of the cabled generators.
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…
We first show that the braid group over a graph topologically containing no $Θ$-shape subgraph has a presentation related only by commutators. Then using discrete Morse theory and triple Massey products, we prove that a graph topologically contains none of four prescribed graphs if and only if its 4-braid groups is a right-angled Artin group.
There are well known relations between braid groups and symmetric groups, between Artin-Briskorn braid groups and Coxeter groups. Inverse braid monoid the same way is related to the inverse symmetric monoid. In the paper we show that similar relations exist between the inverse braid monoid of type $B$ and the inverse reflection monoid of type $B$. This gives a presentation of the last monoid.
Asymptotically rigid mapping class groups III: Presentations and isomorphisms
本文通过群在脊柱复形上的作用，计算了编织Higman-Thompson群的明确展示及其阿贝尔化，并利用这些代数不变量解决了树的渐近刚性映射类群同构问题的多个新情况。
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