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

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。

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。

辫子的n次根在共轭条件下是唯一的

The n-th root of a braid is unique up to conjugacy

我们证明了Makanin的一个猜想：如果a和b是Artin辫群B_n的元素，使得对某个非零整数k，a^k=b^k，则a和b是共轭的.证明涉及Nielsen-Thurston辫子分类。

瑟斯顿之后的映射类群排序

Orderings of mapping class groups after Thurston

本文研究具有非空边界曲面的映射类群。我们提出了一个非常自然的方法，由于瑟斯顿，发现许多不同的左排序等群体。构造方法是在曲面上构造一个双曲结构，在双曲平面上嵌入泛覆盖，并将映射类群对它的作用推广到它在无穷远圆上的极限点.我们对以这种方式产生的辫群的所有序进行分类。此外，限制到一定的类的“nonpathological”序，我们证明了只有1000个共轭类的这种排序。

The level four braid group

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.

Congruence subgroups from representations of the three-strand braid group

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 …

辫群几乎是共同霍普夫式的

Braid groups are almost co-Hopfian

设B_n是n > 3股上的辫群。证明了B_n模其中心是余霍普夫的。证明了B_n的任何单射自同态都是几何的，因为它是由穿孔圆盘的同胚诱导的。进一步证明了从B_n到B_n+1的任意射是几何的。此外，我们得到类似的结果，映射类群的穿孔球。该方法使用瑟斯顿的理论表面同胚和建立在工作的伊万诺夫和麦卡锡。

All principal congruence link groups

We enumerate all the principal congruence link complements in $S^3$, there by answering a question of W. Thurston. Related articles: "Technical Report: All Principal Congruence Link Groups" (arXiv:1902.04722), "All Known Principal Congruence Links" (arXiv:1902.04426).

辫群为线性

Braid groups are linear

在文献[11]中，作者考虑了辫群ρ：B_n \to GL_m（\Bbb Z[q^{ \pm 1}，t^{ \pm 1}]）（m=n（n-1）/2）的一个表示，并证明了它对n=4是忠实的. Bigelow [3]然后通过一个漂亮的拓扑论证证明了相同的表示对所有n都是忠实的。本文给出了对所有n的忠实性的另一种证明。建立了辫群的Charney长度与t的指数之间的关系。构造了模的一个B_n-不变子集，它的性质类似于凸锥的性质.我们将这组中的线段与辫子的Thurston标准形联系起来。

Asymptotic linearity of the mapping class group and a homological version of the Nielsen-Thurston classification

We study the action of the mapping class group on the real homology of finite covers of a topological surface. We use the homological representation of the mapping class to construct a faithful infinite-dimensional representation of the mapping class group. We show that this representation detects the Nielsen-Thurston classification of each mapping class. We then discuss some examples that occur in the theory of braid groups and develop an analogous theory for automorphisms of free groups. We close with some open problems.

小 Coxeter 群的全等子群与晶体学商群

Congruence subgroups and crystallographic quotients of small Coxeter groups

本文研究小 Coxeter 群的全等子群性质，证明无限且非 virtually abelian 的小 Coxeter 群不满足该性质，并确定孪生群与三重群的晶体学商群及其维数和 holonomy 表示。

辫群中字与共轭问题的一种新方法

A new approach to the word and conjugacy problems in the braid groups

研究了$n$ -弦辫群$B_n$的一种新表示.使用它，一个新的解决方案，字问题的$B_n$得到保留的Garside-Thurston解决方案的大部分可取的功能，并在同一时间，使可能的某些计算改进。我们也给出了一个相关的解决共轭问题，但在其复杂性的改善是不清楚在这篇文章中。

Crossing Matrices of Positive Braids

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…

Braids, metallic ratios and periodic solutions of the $2n$-body problem

Periodic solutions of the planar $N$-body problem determine braids through the trajectory of $N$ bodies. Braid types can be used to classify periodic solutions. According to the Nielsen-Thurston classification of surface automorphisms, braids fall into three types: periodic, reducible and pseudo-Anosov. To a braid of pseudo-Anosov type, there is an associated stretch factor greater than 1, and this is a conjugacy invariant of braids. In 2006, the third author discovered a family of multiple choreographic solutions of the planar $2n$-body problem. We prove that braids obtained from the solution…

纤维三维流形的Alexander和Thurston范数

Alexander and Thurston norms of fibered 3-manifolds

对于一个3-流形M，麦克马伦从M的亚历山大多项式导出了一个关于H ^1（M，R）的范数，称为亚历山大范数。他证明了H^1（M，R）上的瑟斯顿范数是亚历山大范数的上界，它度量了对偶曲面的复杂性。他问，当M根纤维在圆上时，这两个范数是否在所有的H^1（M，R）上都相等。在这里，我给的例子表明，这个问题的答案是断然没有。这个问题是有关的忠实性的加斯纳表示的辫群。用来理解这个问题的关键工具是来自组合群论的Bieri-Neumann-Strebel不变量。定理1.7是一个独立的定理，它把Alexander多项式与一个Bieri-Neumann-Strebel不变量联系起来。

Non-realizability of the pure braid group as area-preserving homeomorphisms

Let $\text{Homeo}_+(D^2_n)$ be the group of orientation-preserving homeomorphisms of $D^2$ fixing the boundary pointwise and $n$ marked points as a set. Nielsen realization problem for the braid group asks whether the natural projection $p_n:\text{Homeo}_+(D^2_n)\to B_n:=π_0(\text{Homeo}_+(D^2_n))$ has a section over subgroups of $B_n$. All of the previous methods either use torsions or Thurston stability, which do not apply to the pure braid group $PB_n$, the subgroup of $B_n$ that fixes $n$ marked points pointwise. In this paper, we show that the pure braid group has no realization inside th…

Congruence kernels around affine curves

Let S be a smooth affine algebraic curve, and let S' be the Riemann surface obtained by removing a point from S. We provide evidence for the congruence subgroup property of the mapping class group Mod(S') by showing that its congruence kernel lies in the centralizer of every braid in Mod(S'). As a corollary, we obtain a new proof of Asada's theorem that the congruence subgroup property holds in genus one. We also obtain simple-connectivity of Boggi's procongruence curve complex for curves with at least two punctures, as well as a new proof of Matsumoto's theorem that the congruence kernel depe…

Uniqueness of roots up to conjugacy for some affine and finite type Artin groups

Let $G$ be one of the Artin groups of finite type ${\mathbf B}_n={\mathbf C}_n$, and affine type $\tilde{\mathbf A}_{n-1}$ and $\tilde{\mathbf C}_{n-1}$. In this paper, we show that if $α$ and $β$ are elements of $G$ such that $α^k=β^k$ for some nonzero integer $k$, then $α$ and $β$ are conjugate in $G$. For the Artin group of type $\mathbf A_n$, this was recently proved by J. González-Meneses. In fact, we prove a stronger theorem, from which the above result follows easily by using descriptions of those Artin groups as subgroups of the braid group on $n+1$ strands. Let $P$ be a subset of $\{1…

通过全线性稳定性与对偶辫子群的置换多面体三角剖分

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。

辫子、共形模与熵

Braids, conformal module and entropy

该研究证明辫子共轭类的共形模与熵成反比，可实现两类不变量研究方法的互用，给出希尔伯特第13问题相关定理的短证明，并将共形模用于研究Gromov的Oka原理存在障碍的情形。

辫子群在 $\mathbb{R}$ 上的作用以及低亏格映射类群在 $S^1$ 上作用的刚性

Rigidity of braid group actions on $\mathbb{R}$ and of low-genus mapping class group actions on $S^1$

本文利用辫子群左序空间及其孤立点刻画了 $B_n$ 在 $\mathbb{R}$ 上作用的平移数刚性，并据此将 Mann–Wolff 的 $S^1$ 作用刚性定理推广到带标记点的低亏格曲面。

A generating set for the palindromic Torelli group

A palindrome in a free group F_n is a word on some fixed free basis of F_n that reads the same backwards as forwards. The palindromic automorphism group ΠA_n of the free group F_n consists of automorphisms that take each member of some fixed free basis of F_n to a palindrome; the group ΠA_n has close connections with hyperelliptic mapping class groups, braid groups, congruence subgroups of GL(n,Z), and symmetric automorphisms of free groups. We obtain a generating set for the subgroup of ΠA_n consisting of those elements acting trivially on the abelianisation of F_n, the palindromic Torelli gr…

直角Artin群作为映射类群的正规子群

Right-angled Artin groups as normal subgroups of mapping class groups

本文构造了映射类群中同构于非自由直角Artin群的正规子群的首例，推广了Dahmani-Guirardel-Osin的结果，并给出两个应用：构造不在任何同余子群中的真正规子群及具有特定正规闭包性质的伪Anosov映射类。

Ordered groups and topology

This is a draft of a book submitted for publication by the AMS. Its theme is the remarkable interplay, accelerating in the last few decades, between topology and the theory of orderable groups, with applications in both directions. It begins with an introduction to orderable groups and their algebraic properties. Many of the algebraic results are proved by topological methods, via consideration of the space of orderings. After a discussion Hölder's theorem and some dynamical aspects of orderable groups, we provide explicit orderings of important groups in topology: free groups and most surface…

Quantum computing with Bianchi groups

It has been shown that non-stabilizer eigenstates of permutation gates are appropriate for allowing $d$-dimensional universal quantum computing (uqc) based on minimal informationally complete POVMs. The relevant quantum gates may be built from subgroups of finite index of the modular group $Γ=PSL(2,\mathbb{Z})$ [M. Planat, Entropy 20, 16 (2018)] or more generally from subgroups of fundamental groups of $3$-manifolds [M. Planat, R. Aschheim, M.~M. Amaral and K. Irwin, arXiv 1802.04196(quant-ph)]. In this paper, previous work is encompassed by the use of torsion-free subgroups of Bianchi groups …

Double affine Hecke algebras and congruence groups

The most general construction of double affine Artin groups (DAAG) and Hecke algebras (DAHA) associates such objects to pairs of compatible reductive group data. We show that DAAG/DAHA always admit a faithful action by automorphisms of a finite index subgroup of the Artin group of type $A_{2}$, which descends to a faithful outer action of a congruence subgroup of $SL(2,\mathbb{Z})$ or $PSL(2,\mathbb{Z})$. This was previously known only in some special cases and, to the best of our knowledge, not even conjectured to hold in full generality. The structural intricacies of DAAG/DAHA are captured b…