Artin groups of spherical type up to isomorphism

Right-angled Artin groups and a generalized isomorphism problem for finitely generated subgroups of mapping class groups

Consider the mapping class group $\Mod_{g,p}$ of a surface $Σ_{g,p}$ of genus $g$ with $p$ punctures, and a finite collection $\{f_1,...,f_k\}$ of mapping classes, each of which is either a Dehn twist about a simple closed curve or a pseudo-Anosov homeomorphism supported on a connected subsurface. In this paper we prove that for all sufficiently large $N$, the mapping classes $\{f_1^N,...,f_k^N\}$ generate a right-angled Artin group. The right-angled Artin group which they generate can be determined from the combinatorial topology of the mapping classes themselves. When $\{f_1,...,f_k\}$ are a…

Artin 群的特定结构与同构猜想

A certain structure of Artin groups and the isomorphism conjecture

本文发现一大类 Artin 群的归纳结构，并据此推导出有限实、复和仿射类型 Artin 群的 Farrell-Jones 同构猜想。

Automorphisms and abstract commensurators of 2-dimensional Artin groups

In this paper we consider the class of 2-dimensional Artin groups with connected, large type, triangle-free defining graphs (type CLTTF). We classify these groups up to isomorphism, and describe a generating set for the automorphism group of each such Artin group. In the case where the defining graph has no separating edge or vertex we show that the Artin group is not abstractly commensurable to any other CLTTF Artin group. If, moreover, the defining graph satisfies a further `vertex rigidity' condition, then the abstract commensurator group of the Artin group is isomorphic to its automorphism…

Acylindrical hyperbolicity and Artin-Tits groups of spherical type

We prove that, for any irreducible Artin-Tits group of spherical type $G$, the quotient of $G$ by its center is acylindrically hyperbolic. This is achieved by studying the additional length graph associated to the classical Garside structure on $G$, and constructing a specific element $x_G$ of $G/Z(G)$ whose action on the graph is loxodromic and WPD in the sense of Bestvina-Fujiwara; following Osin, this implies acylindrical hyperbolicity. Finally, we prove that "generic" elements of $G$ act loxodromically, where the word "generic" can be understood in either of the two common usages: as a res…

刻画大类型 Artin 群

Characterising large-type Artin groups

本文通过纯代数刻画证明大类型 Artin 群类在同构下不变，并据此描述其同构类及刚性条件。

Hyperbolic structures for Artin-Tits groups of spherical type

The goal of this mostly expository paper is to present several candidates for hyperbolic structures on irreducible Artin-Tits groups of spherical type and to elucidate some relations between them. Most constructions are algebraic analogues of previously known hyperbolic structures on Artin braid groups coming from natural actions of these groups on curve graphs and (modified) arc graphs of punctured disks.

Basic Questions on Artin-Tits groups

This paper is a short survey on four basic questions on Artin-Tits groups: the torsion, the center, the word problem, and the cohomology ($K(π,1)$ problem). It is also an opportunity to prove three new results concerning these questions: (1) if all free of infinity Artin-Tits groups are torsion free, then all Artin-Tits groups will be torsion free; (2) If all free of infinity irreducible non-spherical type Artin-Tits groups have a trivial center then all irreducible non-spherical type Artin-Tits groups will have a trivial center; (3) if all free of infinity Artin-Tits groups have solutions to …

Admissible submonoids of Artin-Tits monoids

We show the analogue of Muhlherr's [Coxeter groups in Coxeter groups, Finite Geom. and Combinatorics, Cambridge Univ. Press (1993), 277-287] for Artin-Tits monoids, and for Artin-Tits groups of spherical type. That is, the submonoid (resp. subgroup) of an Artin-Tits monoid (resp. group of spherical type) induced by an admissible partition of the Coxeter graph is an Artin-Tits monoid (resp. group). This generalizes and unifies the situation of the submonoid (resp. subgroup) of fixed elements of an Artin-Tits monoid (resp. group of spherical type) under the action of graph automorphisms, and the…

Cryptography with right-angled Artin groups

In this paper we propose right-angled Artin groups as a platform for secret sharing schemes based on the efficiency (linear time) of the word problem. Inspired by previous work of Grigoriev-Shpilrain in the context of graphs, we define two new problems: Subgroup Isomorphism Problem and Group Homomorphism Problem. Based on them, we also propose two new authentication schemes. For right-angled Artin groups, the Group Homomorphism and Graph Homomorphism problems are equivalent, and the later is known to be NP-complete. In the case of the Subgroup Isomorphism problem, we bring some results due to …

On a theorem of Artin, II

This paper is a sequel of [A.M. Cohen, L. Paris, {\it On a theorem of Artin}, J. Group Theory {\bf 6} (2003), 421--441]. Let $A$ be an Artin group, let $W$ be its associated Coxeter group, and let $CA$ be its associated coloured Artin group, that is, the kernel of the standard epimorphism $μ: A \to W$. We determine the homomorphisms $\f: A \to W$ that verify $\Im \f \cdot Z(W)= W$, for $A$ irreducible and of spherical type, and we prove that $CA$ is a characteristic subgroup of $A$, if $A$ is of spherical type but not necessarily irreducible.

球面Artin群的有限商

Finite quotients of spherical Artin groups

本文证明球面和仿射Artin群的最小非交换商与相应Coxeter群的同构，并推出不可约球面Artin群由其有限商群决定。

不可约球型Artin群的最小商与profinite刚性

Smallest quotients and profinite rigidity of irreducible spherical type Artin groups

本文证明不可约球型Artin群在所有球型Artin群中具有profinite刚性，并计算了其最小非交换商，除特殊二面体情形外均与相应Coxeter群一致。

Commensurability in Artin groups of spherical type

Let $A$ and $A'$ be two Artin groups of spherical type, and let $A_1,\dots,A_p$ (resp. $A'_1,\dots,A'_q$) be the irreducible components of $A$ (resp. $A'$). We show that $A$ and $A'$ are commensurable if and only if $p=q$ and, up to permutation of the indices, $A_i$ and $A'_i$ are commensurable for every $i$. We prove that, if two Artin groups of spherical type are commensurable, then they have the same rank. For a fixed $n$, we give a complete classification of the irreducible Artin groups of rank $n$ that are commensurable with the group of type $A_n$. Note that it will remain 6 pairs of gro…

关于有限表现的Bestvina-Brady群的结构

On the structure of finitely presented Bestvina-Brady groups

本文针对Bestvina-Brady群同构型不由定义图唯一确定的问题，证明部分有限表现的该类群可表示为迭代融合积，且该结构可从外围直角Artin群的定义图中读取，丰富了相关群结构的图论刻画。

On the residual nilpotence of pure Artin groups

In this very short note we show that the residual nilpotence of pure Artin groups of spherical type is easily deduced from the faithfulness of the Krammer-Digne representations.

Artin群扭猜想的组合定理

A combination theorem for the twist conjecture for Artin groups

本文将Artin群的强扭猜想简化为定义图无分离顶点的Artin群，提供了新例证并深化了Artin群同构问题的理解。

自由积与标准 Artin 群和对偶 Artin 群之间的同构

Free Products and the Isomorphism between Standard and Dual Artin Groups

本文研究标准 Artin 群与对偶 Artin 群之间满同态 $Ψ$ 为同构的条件，利用非交叉划分偏序集上的 Hurwitz 作用传递性和稳定化子条件给出判定，并证明这些条件在自由积和直积下保持。

The $K(π,1)$-conjecture implies the center conjecture for Artin groups

In this note, we prove that the $K(π,1)$-conjecture for Artin groups implies the center conjecture for Artin groups. Specifically, every Artin group without a spherical factor that satisfies the $K(π,1)$-conjecture has a trivial center.

Normalisateurs et groupes d'Artin-Tits de type sphérique

Let $(A_S,S)$ be an Artin-Tits and $X$ a subset of $S$ ; denote by $A_X$ the subgroup of $A_S$ generated by $X$. When $A_S$ is of spherical type, we prove that the normalizer and the commensurator of $A_X$ in $A_S$ are equal and are the product of $A_X$ by the quasi-centralizer of $A_X$ in $A_S$. Looking to the associated monoids $A_S^+$ and $A_X^+$, we describe the quasi-centralizer of $A_X^+$ in $A_S^+$ thanks to results in Coxeter groups. These two results generalize earlier results of Paris. Finaly, we compare, in the spherical case, the normalizer of a parabolic subgroup in the Artin-Tits…

On a theorem of Artin

We determine the epimorphisms $A \to W$ from the Artin group $A$ of type $Γ$ onto the Coxeter group $W$ of type $Γ$, in case $Γ$ is an irreducible Coxeter graph of spherical type, and we prove that the kernel of the standard epimorphism is a characteristic subgroup of $A$. This generalizes an over 50 years old result of Artin.

On the Subgroups of Right Angled Artin Groups and Mapping Class Groups

There exist right angled Artin groups $A$ such that the isomorphism problem for finitely presented subgroups of $A$ is unsolvable, and for certain finitely presented subgroups the conjugacy and membership problems are unsolvable. It follows that if $S$ is a surface of finite type and the genus of $S$ is sufficiently large, then the corresponding decision problems for the mapping class group $Mod(S)$ are unsolvable. Every virtually special group embeds in the mapping class group of infinitely many closed surfaces. Examples are given of finitely presented subgroups of mapping class groups that h…

On parabolic subgroups of Artin-Tits groups of spherical type

We show that, in an Artin-Tits group of spherical type, the intersection of two parabolic subgroups is a parabolic subgroup. Moreover, we show that the set of parabolic subgroups forms a lattice with respect to inclusion. This extends to all Artin-Tits groups of spherical type a result that was previously known for braid groups. To obtain the above results, we show that every element in an Artin-Tits group of spherical type admits a unique minimal parabolic subgroup containing it. Also, the subgroup associated to an element coincides with the subgroup associated to any of its powers or roots. …

Spherical and geodesic growth rates of right-angled Coxeter and Artin groups are Perron numbers

We prove that for any infinite right-angled Coxeter or Artin group, its spherical and geodesic growth rates (with respect to the standard generating set) either take values in the set of Perron numbers, or equal $1$. Also, we compute the average number of geodesics representing an element of given word length in such groups.

Twisted Lawrence-Krammer representations

Lawrence-Krammer representations are an important family of linear representations of Artin-Tits groups of small type, which are known, under some assumptions on the parameters, to be faithful when the type is spherical (or more generally when they are restricted to the Artin-Tits monoid) and irreducible when the type is connected. Here, we investigate an analogue of these representations --- introduced by Digne in the spherical cases --- for every Artin-Tits monoid that appears as the submonoid of fixed points of an Artin-Tits monoid of small type under a group of graph automorphisms, and for…

扭曲直角Artin群的Droms定理

Droms Theorems for twisted right-angled Artin groups

本文刻画了有限生成子群仍为扭曲直角Artin群的扭曲直角Artin群，基于定义图分类了该类群的凝聚性，并解决了其一个重要子类的同构问题。

Ribbons in Garside monoids

We expound the properties of ribbons in a setting which is general enough to encompass spherical Artin monoids and dual braid monoids of well-generated complex reflection groups. We generalize to our setting results on parabolic subgroups of spherical Artin group of Godelle, Cumplido, and others.

群的图积的同构与自同构

Isomorphisms and automorphisms of graph products of groups

研究群的图积的同构问题，通过给出其自同构群的生成集来解决，此生成集推广了直角阿廷群的相关成果。

Isomorphism and non-isomorphism for interval groups of type D_n

We consider presentations that were derived in \cite{BaumeisterNeaimeRees} for the interval groups associated with proper quasi-Coxeter elements of the Coxeter group $W(D_n)$. We use combinatorial methods to derive alternative presentations for the groups, and use these new presentations to show that the interval group associated with a proper quasi-Coxeter element of $W(D_n)$ cannot be isomorphic to the Artin group of type $D_n$. While the specific problems we solve arise from the study of interval groups, their solution provides an illustration of how techniques indicated by computational ob…

球面Deligne复形中的环及其在Artin群的$K(π,1)$猜想中的应用

Cycles in spherical Deligne complexes and application to $K(π,1)$-conjecture for Artin groups

本研究提出在特定球面Deligne复形中寻找大型非正曲率子复形的方法，用于研究6-环填充，证明了多类Artin群的$K(π,1)$猜想成立，解答了J. McCammond的相关问题。

On the Bieri-Neumann-Strebel-Renz $Σ^1$-invariant of even Artin groups

We calculate the Bieri-Neumann-Strebel-Renz invariant $Σ^1(G)$ for even Artin groups $G$ with underlying graph $Γ$ such that if there is a closed reduced path in $Γ$ with all labels bigger than 2 then the length of such path is always odd. We show that $Σ^1(G)^c$ is a rationally defined spherical polyhedron.

Virtual Artin groups

Starting from the observation that the standard presentation of a virtual braid group mixes the standard presentation of the corresponding braid group with the standard presentation of the corresponding symmetric group and some mixed relations that mimic the action of the symmetric group on its root system, we define a virtual Artin group $VA[Γ]$ of a Coxeter graph $Γ$ mixing the standard presentation of the Artin group $A[Γ]$ with the standard presentation of the Coxeter group $W[Γ]$ and some mixed relations that mimic the action of $W[Γ]$ on its root system. By definition we have two epimorp…

某些球型与仿射型 Artin-Tits 群的 $R_\infty$ 性质

Property $R_\infty$ for some spherical and affine Artin-Tits groups

本文用统一的简短证明方法，证明了多类球型与仿射型 Artin-Tits 群及其纯子群具有 $R_\infty$ 性质，并为纯 Artin 辫子群的已有结论提供了新证明。

On (twisted) Lawrence-Krammer representations

Lawrence-Krammer representations (LK-representations for short) are linear representations of Artin-Tits groups of small type, which are of importance since they are known to be faithful when the type is spherical, or more generally when restricted to the monoid. If the construction is essentially unique for a given small and spherical type, the structure of the set of LK-representations for a given small type is not understood in general. Another important question is to ask if there exists an analogue of this construction in the non-small cases ; a first answer is given in [Digne, On the lin…

Rewriting systems in sufficiently large Artin-Tits groups

A conjecture of Dehornoy claims that, given a presentation of an Artin-Tits group, every word that represents the identity can be transformed into the trivial word using the braid relations, together with certain rules (between pairs of words that are not both positive) that can be derived directly from the braid relations, as well as free reduction, but without introducing trivial factors $ss^{-1} $ or $s^{-1} s$. This conjecture is known to be true for Artin-Tits groups of spherical type or of FC type. We prove the conjecture for Artin--Tits groups of sufficiently large type.

类型(2,3,n)的Artin群

Artin groups of type (2,3,n)

本文证明了类型(2,3,n)的Artin群字问题可通过保长重写规则系统在二次时间内解决，为非球面秩3 Artin群提供了特定的重写解决方案。

具有可视分裂的Artin群的柱形双曲性

Acylindrical hyperbolicity for Artin groups with a visual splitting

该研究建立了判定具有可视分裂的Artin群柱形双曲性的准则，提出抛物子群弱反正规性相关猜想并在多类Artin群中证明，同时建立了相关Artin群猜想间的联系。

类型$B_n$的Artin群的内射映射

Endomorphisms of Artin groups of type $B_n$

研究类型$B_n$的Artin群（n≥5）及其中心商群的内射映射分类，揭示其结构特性。

大型Artin群的外自同构群的VF型

Type VF for outer automorphism groups of large-type Artin groups

本文为连通大型Artin群引入规范形变空间，利用其外自同构群作用，证明了在无三角形或标签至少为6等条件下外自同构群为VF型且有限展示，并推广至具有刚性块的情形。

Right-angled Artin subgroups of Artin groups

The Tits Conjecture, proved by Crisp and Paris, states that squares of the standard generators of any Artin group generate an obvious right-angled Artin subgroup. We consider a larger set of elements consisting of all the centers of the irreducible spherical special subgroups of the Artin group, and conjecture that sufficiently large powers of those elements generate an obvious right-angled Artin subgroup. This alleged right-angled Artin subgroup is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. We verify this conjecture for the class of …

Conjugacy in Artin groups and applications to the classification of surfaces

We show thatthe double reversing algorithm proposed by dehornoy for solving the word problem in the braid group can also be used to recognize the conjugates of powers of the generators in an Artin group of spherical type. The proof uses a characterization of these powers in terms of their fractional decomposition. This algorithm could have potential applications to braid-based cryptography; it also provides a fast method for testing a necessary condition in the classification of surfaces in algebraic geometry.

The codegree isomorphism problem for finite simple groups

We study the codegree isomorphism problem for finite simple groups. In particular, we show that such a group is determined by the codegrees (counting multiplicity) of its irreducible characters. The proof is uniform for all simple groups and only depends on the classification by means of Artin-Tits' simple order theorem.

Bestvina-Brady群的拟阵与同构问题

Matroids and isomorphism problems for Bestvina-Brady groups

本文通过拟阵理论分解Bestvina-Brady群的图同构问题，证明有限表示仅依赖循环拟阵，并构造高连通图使拟阵不同但群同构，且对偶弦图定义的BBG为右角Artin群。

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。

Relative hyperbolicity and Artin groups

This paper considers the question of relative hyperbolicity of an Artin group with regard to the geometry of its associated Deligne complex. We prove that an Artin group is weakly hyperbolic relative to its finite (or spherical) type parabolic subgroups if and only if its Deligne complex is a Gromov hyperbolic space. For a 2-dimensional Artin group the Deligne complex is Gromov hyperbolic precisely when the corresponding Davis complex is Gromov hyperbolic, that is, precisely when the underlying Coxeter group is a hyperbolic group. For Artin groups of FC type we give a sufficient condition for …

无限经典族球面与仿射Artin群的整上同调

Integral cohomology of classical Artin groups: spherical types $A,C,D$ and affine types $\widetilde B,\widetilde C,\widetilde D$

本文利用代数Salvetti复形和Fuchs复形，描述了除仿射族$\widetilde A$外的无限经典族Artin群的整上同调代数加法结构。

类型 $D_5$ 的 Artin 群的自同构

Automorphisms of the Artin group of type $D_5$

本文确定了 $D_5$ 型 Artin 群及其中心商的自同构群，解决了球面型 $D_n$ Artin 群自同构分类中仅剩的 $n=5$ 情形。

Euclidean Artin-Tits groups are acylindrically hyperbolic

In this paper we show the statement in the title. To any Garside group of finite type, Wiest and the author associated a hyperbolic graph called the \emph{additional length graph} and they used it to show that central quotients of Artin-Tits groups of spherical type are acylindrically hyperbolic. In general, a euclidean Artin-Tits group is not \emph{a priori} a Garside group but McCammond and Sulway have shown that it embeds into an \emph{infinite-type} Garside group which they call a \emph{crystallographic Garside group}. We associate a \emph{hyperbolic} additional length graph to this crysta…

Artin群中的固定子群

Fixed subgroups in Artin groups

本文研究大型Artin群自同构的固定子群，定义了自然子群并证明固定子群为有限生成Artin群，秩有界，并给出该子群的几何刻画。

从树到三脚架：关于具有ABI型球面抛物子群的Artin群的K(π,1)证明

From Trees to Tripods: Proof of $K(π,1)$ for Artin groups with $ABI$-type spherical parabolics

本文通过构造注入度量空间的塔，证明了具有特定球面抛物子群性质的Artin群满足K(π,1)猜想。

性质 $R_\infty$ 对新类别的Artin群的性质

Property $R_\infty$ for new classes of Artin groups

本文研究了新类别的Artin群的$R_\infty$性质，通过自同构群和超曲面作用的分析，并提供了Delzant引理的证明。

序、阿廷-蒂茨群、加尔斯德结构

Ordering, Artin--Tits group, Garside structure

本文研究球形型阿廷群的德霍诺伊结构相关条件，证明其A条件几乎总是成立但B条件很少满足，明确了这类群具备德霍诺伊结构的条件情况。

The conjugacy stability problem for parabolic subgroups in Artin groups

Given an Artin group $A$ and a parabolic subgroup $P$, we study if every two elements of $P$ that are conjugate in $A$, are also conjugate in $P$. We provide an algorithm to solve this decision problem if $A$ satisfies three properties that are conjectured to be true for every Artin group. We partially solve the problem if $A$ has $FC$-type, and we totally solve it if $A$ is isomorphic to a free product of spherical Artin groups. In particular, we show that in this latter case, every element of $A$ is contained in a unique minimal (by inclusion) parabolic subgroup.

球型阿廷群的$K(π, 1)$猜想

The $K(π, 1)$ conjecture for Artin groups of spherical type

介绍球型阿廷群的$K(π, 1)$猜想，目标是通过组合拓扑方法，在只移除有限个超平面的球型情形下证明该猜想，其证明灵感源于相关原始证明。

球形Bestvina-Brady核的欧拉示性数

The Euler Characteristic of Spherical Bestvina-Brady Kernels

该研究计算所有球形型Artin群的Bestvina-Brady核的欧拉示性数，利用离散莫尔斯理论分析其分类空间结构，推广计算方法至满足特定条件的Artin群并给出实例。