Paolo Bellingeri, Eddy Godelle
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.
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We prove that pure braid groups of closed surface are almost-direct products of residually torsion free nilpotent groups and hence residually torsion free nilpotent. As a Corollary, we prove also that braid groups on 2 strands of closed surfaces are residually nilpotent.
We prove by explicit construction that graph braid groups and most surface groups can be embedded in a natural way in right-angled Artin groups, and we point out some consequences of these embedding results. We also show that every right-angled Artin group can be embedded in a pure surface braid group. On the other hand, by generalising to right-angled Artin groups a result of Lyndon for free groups, we show that the Euler characteristic -1 surface group (given by the relation x^2y^2=z^2) never embeds in a right-angled Artin group.
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.
We define and study extensions of Artin's representation and braid monodromy representation to the case of topological and algebraical generalisations of braid groups. In particular we provide faithful representations of braid groups of oriented surfaces with boundary components as (outer) automorphisms of free groups. We give also similar representations for braid groups of non oriented surfaces with boundary components and we show a representation of braid groups of closed surfaces as outer automorphisms of free groups. Finally, we provide faithful representations of Artin-Tits groups of typ…
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。
Herein we prove that if $M$ is a compact oriented Riemann surface of genus $g$, and $M^{[n]}$ is the classifying space of $n$ distinct, unordered points on $M$, then the kernel of the map $π_1(M^{[n]})\to H_1(M)$ is generated by transpositions for sufficiently large $n$. Specifically, we treat $M$ as a polyhedron, and the edge set of $M$ generates this group.
We determine the lower central series and corresponding residual properties for braid groups and pure braid groups of orientable surfaces.
We prove that the pure braid groups on closed, orientable surfaces are bi-orderable, and that the pure braid groups on closed, non-orientable surfaces have generalized torsion, thus they are not bi-orderable.
We show that Vassiliev invariants separate braids on a closed oriented surface, and we exhibit an universal Vassiliev invariant for these braids in terms of chord diagrams labeled by elements of the fundamental group of the considered surface.
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.
In this paper, we prove that each automorphism of a surface braid group is induced by a homeomorphism of the underlying surface, provided that this surface is a closed, connected, orientable surface of genus at least 2, and the number of strings is at least three. This result generalizes previous results for classical braid groups, mapping class groups, and Torelli groups.
Let $M$ be a closed oriented surface of genus $g\ge 1$, let $B_n(M)$ be the braid group of $M$ on $n$ strings, and let $SB_n(M)$ be the corresponding singular braid monoid. Our purpose in this paper is to prove that the desingularization map $η: SB_n(M) \to \Z [B_n(M)]$, introduced in the definition of the Vassiliev invariants (for braids on surfaces), is injective.
We propose a family of new representations of the braid groups on surfaces that extend linear representations of the braid groups on a disc such as the Burau representation and the Lawrence-Krammer-Bigelow representation.
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.
We consider exact sequences and lower central series of surface braid groups and we explain how they can prove to be useful for obtaining representations for surface braid groups. In particular, using a completely algebraic framework, we describe the notion of extension of a representation introduced and studied recently by An and Ko and independently by Blanchet.
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.
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.
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…
We determine a set of generators for the Brunnian braids on a general surface $M$ for $M\not=S^2$ or $\RP^2$. For the case $M=S^2$ or $\RP^2$, a set of generators for the Brunnian braids on $M$ is given by our generating set together with the homotopy groups of a 2-sphere.
In [J.Birman, V.Gebhardt, J.Gonzalez-Meneses, Conjugacy in Garside groups I: cyclings, powers and rigidity] authors asked: (open question 2) is the size of USS of a rigid pseudo-Anosov braid is bounded above by some polynomial in the number of strands and the braid length? We answer this question in the negative.
Asymptotic and cohomological dimension of surface braid groups and poly-surface groups
本文确定了所有曲面辫群（包括非定向和无限类型曲面相关群）的渐近维数，以及无 torsion 多有限生成曲面群的渐近维数。证明了这两类群的虚拟上同调维数与渐近维数相同，并展示了这些群在 Bieri-Eckmann 意义下是虚拟对偶群。
We prove that there is no functorial universal finite type invariant for braids in $Σ\times I$ if the genus of $Σ$ is positive.
We prove that if g and n are integers at least two, then the abstract commensurator of the braid group with n strands on a closed orientable surface of genus g is naturally isomorphic to the extended mapping class group of a compact orientable surface of genus g with n boundary components.
The closure of a braid in a closed orientable surface $Σ$ is a link in $Σ\times S^1$. We classify such closed surface braids up to isotopy and homeomorphism (with a small indeterminacy for isotopy of closed sphere braids), algebraically in terms of the surface braid group. We find that in positive genus, braids close to isotopic links if and only if they are conjugate, and close to homeomorphic links if and only if they are in the same orbit of the outer action of the mapping class group on the surface braid group modulo its center.
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.
We construct a universal Vassiliev invariant for braid groups of the sphere and the mapping class groups of the sphere with $n$ punctures. The case of a sphere is different from the classical braid groups or braids of oriented surfaces of genus strictly greater than zero, since Vassiliev invariants in a group without 2-torsion do not distinguish elements of braid group of a sphere.
Lower central series, surface braid groups, surjections and permutations
本文推广了Artin和Lin关于经典辫群的结果，确定了无边界可定向曲面辫群间满射存在的条件，并给出了有边界及不可定向曲面的部分结果及对称群表示分类。
In this paper we construct a gathering process by the means of which we obtain new normal forms in braid groups. The new normal forms generalise Artin-Markoff normal forms and possess an extremely natural geometric description. In the two last sections of the paper we discuss the implementation of the introduced gathering process and the questions that arose in our work. This discussion leads us to some interesting observations, in particular, we offer a method of generating a random braid.
Braid combing is a procedure defined by Emil Artin to solve the word problem in braid groups for the first time. It is well-known to have exponential complexity. In this paper, we use the theory of straight line programs to give a polynomial algorithm which performs braid combing. This procedure can be applied to braids on surfaces, providing the first algorithm (to our knowledge) which solves the word problem for braid groups on surfaces with boundary in polynomial time and space. In the case of surfaces without boundary, braid combing needs to use a section from the fundamental group of the …
A four-term exact sequence of surface orbifold pure braid groups
本文证明了亏格大于等于1且带锥点的二维可定向曲面轨形纯辫群存在四项正合序列，并纠正了此前arXiv.2106.08110中的结果。
Let g and n be integers at least two, and let G be the pure braid group with n strands on a closed orientable surface of genus g. We describe any injective homomorphism from a finite index subgroup of G into G. As a consequence, we show that any finite index subgroup of G is co-Hopfian.
This paper is a survey of some properties of the braid groups and related groups that lead to questions on mapping class groups.
We show that the morphisms from the braid group with n strands in the mapping class group of a surface with a possible non empty boundary, assuming that its genus is smaller or equal to n/2 are either cyclic morphisms (their images are cyclic groups), or transvections of monodromy morphisms (up to multiplication by an element in the centralizer of the image, the image of a standard generator of the braid group is a Dehn twist, and the images of two consecutive standard generators are two Dehn twists along two curves intersecting in one point). As a corollary, we determine the endomorphisms, th…
Smallest nonabelian quotients of surface braid groups
本文给出曲面辫群非交换商大小的精确下界，并分类达到下界的商为对称群或2步幂零p-群。
The R$_{\infty}$-property for braid groups over orientable surfaces
本文研究了可定向曲面上纯辫群与全辫群的 R$_{\infty}$-性质，并证明除少数例外，这些群均具有该性质。
Automorphism group of the commutator subgroup of the braid group
本文研究辫子群交换子子群$B'_n$的自同构群，证明$n\ge4$时其自同构群等于$B_n$的自同构群，解决了Vladimir Lin的问题。
Braided surfaces and their characteristic maps
本文研究曲面分支覆盖何时成为编织曲面的特征映射，利用经自由群与辫群的单值分解给出构造，并以辫群 Hermitian 表示刻画其强等价类的有限完备可区分性。
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.
We give a survey of the theory of surface braid groups and the lower algebraic K-theory of their group rings. We recall several definitions and describe various properties of surface braid groups, such as the existence of torsion, orderability, linearity, and their relation both with mapping class groups and with the homotopy groups of the 2-sphere. The braid groups of the 2-sphere and the real projective plane are of particular interest because they possess elements of finite order, and we discuss in detail their torsion and the classification of their finite and virtually cyclic subgroups. F…
We study the minimal dilatation of pseudo-Anosov pure surface braids and provide upper and lower bounds as a function of genus and the number of punctures. For a fixed number of punctures, these bounds tend to infinity as the genus does. We also bound the dilatation of pseudo-Anosov pure surface braids away from zero and give a constant upper bound in the case of a sufficient number of punctures.
Pure braid groups are RFRS
该研究证明纯辫群是RFRS，肯定了Agol2014年提出的辫群是否为虚拟RFRS的问题，还为Artin群是否虚拟特殊的问题提供了证据。
Coxeter-type quotients of surface braid groups
本文研究了曲面辫群在元素 $σ_1^q$ 正规闭包下的 Coxeter 型商群，以及闭可定向曲面或圆盘上结合纯辫群换位子群的 Coxeter 型商群。
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.
For every group genetic code with finite number of generating and at most with one defining relation we introduce the braid group of this genetic code. This construction includes the braid group of Euclidean plane, the braid groups of closed orientable surfaces, B type groups of Artin-Brieskorn, and allow us to study all these groups with common point of view. We present some results on the structure of the braid groups of genetic codes, describe normal form of words, torsion, and some results on widths of the verbal subgroups. Also we prove that Scott's system of defining relations for the br…
Let M be a compact surface, either orientable or non-orientable. We study the lower central and derived series of the braid and pure braid groups of M in order to determine the values of n for which B\_n(M) and P\_n(M) are residually nilpotent or residually soluble. First, we solve this problem for the case where M is the 2-torus. We then give a general description of these series for an arbitrary semi-direct product that allows us to calculate explicitly the lower central series of P\_2(K), where K is the Klein bottle, and to give an estimate for the derived series of P\_n(K). Finally, if M i…
We show that several important normal subgroups $Γ$ of the mapping class group of a surface satisfy the following property: any free, ergodic, probability measure preserving action $Γ\curvearrowright X$ is stably OE-superrigid. These include the central quotients of most surface braid groups and most Torelli groups and Johnson kernels. In addition, we show that all these groups satisfy the measure equivalence rigidity and we describe all their lattice-embeddings. Using these results in combination with previous results from [CIK13] we deduce that any free, ergodic, probability measure preservi…
Graph of groups decompositions of graph braid groups
本文提供了一种显式构造将图形辫群分解为群图的方法，计算了多种图形的辫群，回答了Genevois的多个问题，并区分了直角Artin群与图形辫群。
For a general surface $M$ and an arbitrary braid $α$ from the surface braid group $B_{n-1}(M)$ we study the system of equations $d_1β=\cdots=d_{n}β=α, $ where operation $d_i$ is deleting of $i$-th strand. We obtain that if $M\not=S^2$ or $\mathbb RP^2$ this system of equations has a solution $β\in B_{n}(M)$ if and only if $d_1α=\ldots=d_nα. $ The set of braids satisfying the last system of equations we call Cohen braids. We also construct a set of generators for the groups of Cohen braids. In the cases of the sphere and the projective plane we give some examples for the small number of strands…
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.
This article is an exposition of certain connections between the braid groups, classical homotopy groups of the 2-sphere, as well as Lie algebras attached to the descending central series of pure braid groups arising as Vassiliev invariants of pure braids. Natural related questions are posed at the end of this article.
Let M be a compact, connected non-orientable surface without boundary and of genus g greater than or equal to 3. We investigate the pure braid groups P_n(M) of M, and in particular the possible splitting of the Fadell-Neuwirth short exact sequence 1 --> P_m(M {x_1,...,x_n}) --> P_{n+m}(M) --> P_n(M) --> 1, where m,n are positive integers, and the homomorphism p*:P_{n+m}(M) --> P_n(M) corresponds geometrically to forgetting the last m strings. This problem is equivalent to that of the existence of a section for the associated fibration p:F_{n+m}(M)} --> F_n(M) of configuration spaces, defined b…
In this paper we introduce a new class of non-amenable groups denoted by ${\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ which give rise to $\textit{prime}$ von Neumann algebras. This means that for every $Γ\in {\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ its group von Neumann algebra $L(Γ)$ cannot be decomposed as a tensor product of diffuse von Neumann algebras. We show ${\bf NC}_1 \cap {\bf Quot}(\mathcal C_{rss})$ is fairly large as it contains many examples of groups intensively studied in various areas of mathematics, notably: all infinite central quotients of pure surface braid groups; …
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。
On finite quotients of surface braid groups having order at most $127$
本文研究了曲面辫群的有限商，分类了所有阶数不超过127的可接受商。
We consider finite-sheeted, regular, possibly branched covering spaces of compact surfaces with boundary and the associated liftable and symmetric mapping class groups. In particular, we classify when either of these subgroups coincides with the entire mapping class group of the surface. As a consequence, we construct infinite families of non-geometric embeddings of the braid group into mapping class groups in the sense of Wajnryb. Indeed, our embeddings map standard braid generators to products of Dehn twists about curves forming chains of arbitrary length. As key tools, we use the Birman-Hil…
In this paper we first show that many braid groups of low genus surfaces have their centers as direct factors. We then give a description of centralizers and normalizers of prime order elements in pure mapping class groups of surfaces with spherical quotients using automorphism groups of fundamental groups of the quotient surfaces. As an application, we use these to show that the $p$-primary part of the Farrell cohomology groups of certain mapping class groups are elementary abelian groups. At the end we compute the $p$-primary part of the Farrell cohomology of a few pure mapping class groups.
Finite quotients of surface braid groups and double Kodaira fibrations
该研究阐述了近期关于两弦纯辫群$\mathsf{P}_2(\Sigma_b)$的非阿贝尔有限商（纯辫商）的相关成果，并介绍了利用这类群构建新双重小平纤维化的方法。
Finite quotients of full surface braid groups and complex surfaces of general type: cyclic, dihedral, and extra-special quotients
研究亏格为\(g\)的紧致黎曼曲面上双股全辫群的有限循环、二面体和超特殊商群数量，在商映射特定条件下计算。并将代数结果用于构造一般型光滑曲面，得到两个三维族的极小曲面，其有相同双正则不变量和贝蒂数，但第一同调群挠部分不同。
We prove that the symplectic group $Sp(2n,\mathbb Z)$ and the mapping class group $Mod_{S}$ of a compact surface $S$ satisfy the $R_{\infty}$ property. We also show that $B_n(S)$, the full braid group on $n$-strings of a surface $S$, satisfies the $R_{\infty}$ property in the cases where $S$ is either the compact disk $D$, or the sphere $S^2$. This means that for any automorphism $ϕ$ of $G$, where $G$ is one of the above groups, the number of twisted $ϕ$-conjugacy classes is infinite.
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…
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