A Poset Connected to Artin Monoids of Simply Laced Type

Parabolic subgroups of large-type Artin groups

We show that the geometric realisation of the poset of proper parabolic subgroups of a large-type Artin group has a systolic geometry. We use this geometry to show that the set of parabolic subgroups of a large-type Artin group is stable under arbitrary intersections and forms a lattice for the inclusion. As an application, we show that parabolic subgroups of large-type Artin groups are stable under taking roots and we completely characterise the parabolic subgroups that are conjugacy stable. We also use this geometric perspective to recover and unify results describing the normalisers of para…

Root systems, symmetries and linear representations of Artin groups

Let $Γ$ be a Coxeter graph, let $W$ be its associated Coxeter group, and let $G$ be a group of symmetries of $Γ$.Recall that, by a theorem of H{é}e and Mühlherr, $W^G$ is a Coxeter group associated to some Coxeter graph $\hat Γ$.We denote by $Φ^+$ the set of positive roots of $Γ$ and by $\hat Φ^+$ the set of positive roots of $\hat Γ$.Let $E$ be a vector space over a field $\K$ having a basis in one-to-one correspondence with $Φ^+$.The action of $G$ on $Γ$ induces an action of $G$ on $Φ^+$, and therefore on $E$.We show that $E^G$ contains a linearly independent family of vectors naturally in o…

Dual euclidean Artin groups and the failure of the lattice property

The irreducible euclidean Coxeter groups that naturally act geometrically on euclidean space are classified by the well-known extended Dynkin diagrams and these diagrams also encode the modified presentations that define the irreducible euclidean Artin groups. These Artin groups have remained mysterious with some exceptions until very recently. Craig Squier clarified the structure of the three examples with three generators more than twenty years ago and François Digne more recently proved that two of the infinite families can be understood by constructing a dual presentation for each of these…

仿射 Artin 群的 $K(\pi, 1)$ 猜想

The $K(π, 1)$ conjecture for affine Artin groups

本文综述仿射 Artin 群 $K(\pi, 1)$ 猜想近期证明的核心思路，从非交叉划分偏序集、dual Artin 群及拓扑模型三方面梳理论证。

秩为3的Coxeter群与Artin群上的对偶结构

Dual structures on Coxeter and Artin groups of rank three

本研究将对偶Coxeter群与Artin群理论拓展至全部秩3 Coxeter系统，证明相关非交叉分拆偏序集的性质及秩3 Artin群的多个猜想，部分构造可推广至一般Artin群，为领域公开问题的解决提供新思路。

Weighted sheaves and homology of Artin groups

In this paper we expand the theory of weighted sheaves over posets, and use it to study the local homology of Artin groups. First, we use such theory to relate the homology of classical braid groups with the homology of certain independence complexes of graphs. Then, in the context of discrete Morse theory on weighted sheaves, we introduce a particular class of acyclic matchings. Explicit formulas for the homology of the corresponding Morse complexes are given, in terms of the ranks of the associated incidence matrices. We use such method to perform explicit computations for the new affine cas…

右角Artin群纯对称外自同构群的上同调

Cohomology of the pure symmetric automorphisms of right-angled Artin groups

本文计算了右角Artin群A_Γ的纯对称外自同构群ΣPOut(A_Γ)和纯对称自同构群ΣPAut(A_Γ)的上同调群，利用等变谱序列和Lyndon-Hochschild-Serre谱序列，证明了上同调环由度1生成，并提出广义Brownstein-Lee猜想。

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。

连续非交叉划分与加权循环分解

Continuous Noncrossing Partitions and Weighted Circular Factorizations

本文研究了复平面单位圆上的连续非交叉划分，证明其构成拓扑偏序集并同胚于辫群的对偶Garside分类空间，并将此推广至任意Coxeter群中基于Coxeter元的加权线性分解构造。

扩张 Weyl 群、Hurwitz 传递性与加权射影线 I：一般理论与管状情形

Extended Weyl groups, Hurwitz transitivity and weighted projective lines I: Generalities and the tubular case

本文研究扩张 Weyl 群与加权射影线上厚子范畴的组合对应，证明管状类型下 Hurwitz 作用在 Coxeter 元约化反射分解上的传递性，从而建立保序双射。

有向图、Frattini抗性与极大pro-$p$伽罗瓦群

Directed graphs, Frattini-resistance, and maximal pro-$p$ Galois groups

该研究针对定向右角Artin pro-$p$群，证明了其关联有向图为初等型、具有Frattini抗性、拓扑有限生成闭子群均为定向pro-$p$ RAAG、是含$p$次单位根域的极大pro-$p$伽罗瓦群这四个条件等价，并提出相关上同调无本质三重Massey积的猜想。

通过初等工具访问Grothendieck-Teichmueller群的非交换商

Accessing non-abelian quotients of the Grothendieck-Teichmueller group via elementary tools

本文通过构造有限非交换商群和profinite商群，证明了从绝对伽罗华群到Grothendieck-Teichmueller群的满射性。

奇数 D 型的有理 Weyl 群元素

Rational Weyl group elements of odd type D

本文证明了奇数 r 的 D_r 型有理 Weyl 群元素恰好是长元素 w_0 和每个非空子集 I 对应的两个符号循环元素 c_I 和 d_I，从而验证了 Voloshyn 的猜想并给出了有理图的结构。