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

Extra-large type Artin groups are hierarchically hyperbolic

We show that Artin groups of extra-large type, and more generally Artin groups of large and hyperbolic type, are hierarchically hyperbolic. This implies in particular that these groups have finite asymptotic dimension and uniform exponential growth. We prove these results by using a combinatorial approach to hierarchical hyperbolicity, via the action of these groups on a new complex that is quasi-isometric both to the coned-off Deligne complex introduced by Martin-Przytycki and to a generalisation due to Morris-Wright of the graph of irreducible parabolic subgroups of finite type introduced by…

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。

The structure of euclidean Artin groups

The Coxeter groups that act geometrically on euclidean space have long been classified and presentations for the irreducible ones are encoded in the well-known extended Dynkin diagrams. The corresponding Artin groups are called euclidean Artin groups and, despite what one might naively expect, most of them have remained fundamentally mysterious for more than forty years. Recently, my coauthors and I have resolved several long-standing conjectures about these groups, proving for the first time that every irreducible euclidean Artin group is a torsion-free centerless group with a decidable word …

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。

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…

Artin groups of euclidean type

This article resolves several long-standing conjectures about Artin groups of euclidean type. In particular, we prove that every irreducible euclidean Artin group is a torsion-free centerless group with a decidable word problem and a finite-dimensional classifying space. We do this by showing that each of these groups is isomorphic to a subgroup of a group with an infinite-type Garside structure. The Garside groups involved are introduced here for the first time. They are constructed by applying semi-standard procedures to crystallographic groups that contain euclidean Coxeter groups but which…

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。

上同调环与偶Artin群的$p$-局部行为

Cohomology rings and $p$-local behavior of even Artin groups

将右角Artin群的经典结果推广到偶Artin群族，计算了FC型偶Artin群的上同调环、pro-$p$完备化和$p$-Zassenhaus限制李代数，并证明了图$p$部分的刚性。

On the local homology of Artin groups of finite and affine type

We study the local homology of Artin groups using weighted discrete Morse theory. In all finite and affine cases, we are able to construct Morse matchings of a special type (we call them "precise matchings"). The existence of precise matchings implies that the homology has a square-free torsion. This property was known for Artin groups of finite type, but not in general for Artin groups of affine type. We also use the constructed matchings to compute the local homology in all exceptional cases, correcting some results in the literature.

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 …

A Garside presentation for Artin-Tits groups of type $\tilde C_n$

We prove that an Artin-Tits group of type $\tilde C$ is the group of fractions of a Garside monoid, analogous to the known dual monoids associated with Artin-Tits groups of spherical type and obtained by the "generated group" method. This answers, in this particular case, a general question on Artin-Tits groups, gives a new presentation of an Artin-Tits group of type $\tilde C$, and has consequences for the word problem, the computation of some centralizers or the triviality of the center. A key point of the proof is to show that this group is a group of fixed points in an Artin-Tits group of …

二面体 Artin 群中的扭曲共轭 II：Baumslag–Solitar 群 $\mathrm{BS}(n,n)$

Twisted conjugacy in dihedral Artin groups II: Baumslag Solitar groups $\mathrm{BS}(n,n)$

本文针对偶数二面体 Artin 群解决扭曲共轭问题，并通过证明所有自同构子群的轨道可判定性，确立其扩张中共轭问题的可解性。

关于FC型偶阿廷群的Sigma不变量

On the Sigma invariants of even Artin groups of FC-type

本文研究FC型偶阿廷群的Sigma不变量，提出强同调n-连通条件作为相关特征的充分条件，给出同调群自由部分的公式并修正了此前版本的两处错误。

Injections of Artin groups

We study those Artin groups which, modulo their centers, are finite index subgroups of the mapping class group of a sphere with at least 5 punctures. In particular, we show that any injective homomorphism between these groups is parameterized by a homeomorphism of a punctured sphere together with a map to the integers. We also give a generating set for the automorphism group of the pure braid group on at least 4 strands. The technique, following Ivanov, is to prove that every superinjective map of the complex of curves of a sphere with at least 5 punctures is induced by a homeomorphism.

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…

On virtual indicability and property (T) for outer automorphism groups of RAAGs

We give a condition on the defining graph of a right-angled Artin group which implies its automorphism group is virtually indicable, that is, it has a finite-index subgroup that admits a homomorphism onto $\Z$. We use this as part of a criterion that determines precisely when the outer automorphism group of a right-angled Artin group defined on a graph with no separating intersection of links has property (T). As a consequence we also obtain a similar criterion for graphs in which each equivalence class under the domination relation of Servatius generates an abelian group.

Artin groups of types $F_4$ and $H_4$ are not commensurable with that of type $D_4$

In a recent article, Cumplido and Paris studied the question of commensurability between Artin groups of spherical type. Their analysis left six cases undecided, for the following pairs of Artin groups: $(F_4,D_4)$, $(H_4,D_4)$, $(F_4,H_4)$, $(E_6,D_6)$, $(E_7,D_7)$, and $(E_8,D_8)$. In this note we resolve the first two of these cases, namely, we show that the Artin groups of types $F_4$ and $H_4$ are not commensurable with that of type $D_4$. As a key step, we realize the abstract commensurator of the Artin group of type $D_4$ as the extended mapping class group of the torus with three punct…

关于 $D_n$ 型球面 Deligne 复形

On spherical Deligne complexes of type $D_n$

本文研究 $D_n$ 型 Artin 群的球面 Deligne 复形，证明其 1 骨架中特定 6-环具有中心或拟中心，并推出某二维相对 Artin 复形在 Moussong 度量下为 CAT$(1)$。

Poly-freeness of Artin groups and the Farrell-Jones Conjecture

We provide two simple proofs of the fact that even Artin groups of FC-type are poly-free which was recently established by R. Blasco-Garcia, C. Martínez-Pérez and L. Paris. More generally, let $Γ$ be a finite simplicial graph with all edges labelled by positive even integers and $A_Γ$ be its associated Artin group; our new proof implies that if $A_T$ is poly-free (resp. normally poly-free) for every clique $T$ in $Γ$, then $A_Γ$ is poly-free (resp. normally poly-free). We prove similar results regarding the Farrell--Jones Conjecture for even Artin groups. In particular, we show that if $A_Γ$ i…

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。

有限型 Artin 群的标记图

A marking graph for finite-type Artin groups

本文为不可约有限型 Artin 群构造了与其模中心商空间拟等距的标记图，通过横截抛物子群复形扩展极大单纯形并定义初等移动的类比，推广了 Masur-Minsky 层级机制。

Cactus groups, twin groups, and right-angled Artin groups

Cactus groups Jn are currently attracting considerable interest from diverse mathematical communities. This work explores their relations to right-angled Coxeter groups, and in particular twin groups Twn and Mostovoy's Gauss diagram groups Dn, which are better understood. Concretely, we construct an injective group 1-cocycle from Jn to Dn, and show that Twn (and its k-leaf generalisations) inject into Jn. As a corollary, we solve the word problem for cactus groups, determine their torsion (which is only even) and center (which is trivial), and answer the same questions for pure cactus groups, …

Orbit equivalence rigidity of irreducible actions of right-angled Artin groups

Let $G_Γ\curvearrowright X$ and $G_Λ\curvearrowright Y$ be two free measure-preserving actions of one-ended right-angled Artin groups with trivial center on standard probability spaces. Assume they are irreducible, i.e. every element from a standard generating set acts ergodically. We prove that if the two actions are stably orbit equivalent (or merely stably $W^*$-equivalent), then they are automatically conjugate through a group isomorphism between $G_Γ$ and $G_Λ$. Through work of Monod and Shalom, we derive a superrigidity statement: if the action $G_Γ\curvearrowright X$ is stably orbit equ…

关于正规多自由群的虚拟循环维数

On the virtually cyclic dimension of normally poly-free groups

本文给出了正规多自由群虚拟循环维数关于长度的上界，并由此证明FC型虚拟偶Artin群存在有限维分类空间模型。

Helly meets Garside and Artin

A graph is Helly if every family of pairwise intersecting combinatorial balls has a nonempty intersection. We show that weak Garside groups of finite type and FC-type Artin groups are Helly, that is, they act geometrically on Helly graphs. In particular, such groups act geometrically on spaces with convex geodesic bicombing, equipping them with a nonpositive-curvature-like structure. That structure has many properties of a CAT(0) structure and, additionally, it has a combinatorial flavor implying biautomaticity. As immediate consequences we obtain new results for FC-type Artin groups (in parti…

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…

Bestvina's normal form complex and the homology of Garside groups

A Garside group is a group admitting a finite lattice generating set D. Using techniques developed by Bestvina for Artin groups of finite type, we construct K(π,1)s for Garside groups. This construction shows that the (co)homology of any Garside group G is easily computed given the lattice D, and there is a simple sufficient condition that implies G is a duality group. The universal covers of these K(π,1)s enjoy Bestvina's weak non-positive curvature condition. Under a certain tameness condition, this implies that every solvable subgroup of G is virtually abelian.

Malnormality and centers in one-relator relative presentations

Let H be a group obtained from a group G by adding one generator and one relator which is a proper power of a unimodular word. The main aim of this paper is to prove that G is a malnormal subgroup of H. This implies that if G is a nontrivial group, then the group H has a trivial center.

温和的p-群与有序独异点

Mild Pro-$p$ Groups and Ordered Monoids

本文提出一个有限呈现实现p-群温和性的判据，推广了Schmidt和Gärtner的结果，并涵盖了Labute的非奇异电路判据，同时与Quadrelli等人关于p-右角Artin群温和性的三角条件相联系。

Generalized small cancellation conditions, non-positive curvature and diagrammatic reducibility

We present a metric condition ${\LARGEτ}'$ which describes the geometry of classical small cancellation groups and applies also to other known classes of groups such as two-dimensional Artin groups. We prove that presentations satisfying condition ${\LARGEτ}'$ are diagrammatically reducible in the sense of Sieradski and Gersten. In particular we deduce that the standard presentation of an Artin group is aspherical if and only if it is diagrammatically reducible. We show that, under some extra hypotheses, ${\LARGEτ}'$-groups have quadratic Dehn functions and solvable conjugacy problem. In the s…

On the center of a Coxeter group

In this paper, we show that the center of every Coxeter group is finite and isomorphic to $(\Z_2)^n$ for some $n\ge 0$. Moreover, for a Coxeter system $(W,S)$, we prove that $Z(W)=Z(W_{S\setminus\tilde{S}})$ and $Z(W_{\tilde{S}})=1$, where $Z(W)$ is the center of the Coxeter group $W$ and $\tilde{S}$ is the subset of $S$ such that the parabolic subgroup $W_{\tilde{S}}$ is the {\it essential parabolic subgroup} of $(W,S)$ (i.e.\ $W_{\tilde{S}}$ is the minimum parabolic subgroup of finite index in $(W,S)$). The finiteness of the center of a Coxeter group implies that a splitting theorem holds fo…

Conjugacy in Garside Groups III: Periodic braids

An element in Artin's braid group B_n is said to be periodic if some power of it lies in the center of B_n. In this paper we prove that all previously known algorithms for solving the conjugacy search problem in B_n are exponential in the braid index n for the special case of periodic braids. We overcome this difficulty by putting to work several known isomorphisms between Garside structures in the braid group B_n and other Garside groups. This allows us to obtain a polynomial solution to the original problem in the spirit of the previously known algorithms. This paper is the third in a series…

Determining Fuchsian groups by their finite quotients

Let $\C(Γ)$ be the set of isomorphism classes of the finite groups that are homomorphic images of $Γ$. We investigate the extent to which $\C(Γ)$ determines $Γ$ when $Γ$ is a group of geometric interest. If $Γ_1$ is a lattice in ${\rm{PSL}}(2,\R)$ and $Γ_2$ is a lattice in any connected Lie group, then $\C(Γ_1) = \C(Γ_2)$ implies that $Γ_1$ is isomorphic to $Γ_2$. If $F$ is a free group and $Γ$ is a right-angled Artin group or a residually free group (with one extra condition), then $\C(F)=\C(Γ)$ implies that $F\congΓ$. If $Γ_1<{\rm{PSL}}(2,\Bbb C)$ and $Γ_2< G$ are non-uniform arithmetic latt…

Orbifold 辫群

Orbifold braid groups

本文基于先前对 orbifold 辫群的研究，证明了 orbifold 的一类一般子 orbifold 包含映射诱导的单射同态，并得出大多数 orbifold 辫群的中心是平凡的结论。

Finite group extensions and the Atiyah conjecture

The Atiyah conjecture for a discrete group G states that the $L^2$-Betti numbers of a finite CW-complex with fundamental group G are integers if G is torsion-free and are rational with denominators determined by the finite subgroups of G in general. Here we establish conditions under which the Atiyah conjecture for a group G implies the Atiyah conjecture for every finite extension of G. The most important requirement is that the cohomology $H^*(G,\mathbb{Z}/p)$ is isomorphic to the cohomology of the p-adic completion of G for every prime p. An additional assumption is necessary, e.g. that the …

具有循环中心的幂零类2群的模同构问题研究

On the modular isomorphism problem for groups of nilpotency class 2 with cyclic center

该研究解决了具有循环中心的幂零类2群的模同构问题，证明p元域下这类p-群的群代数同构可推出群同构，奇数阶群在任意特征p域下也成立，偶数阶群需附加群或域的假设。

Basic results on braid groups

These are Lecture Notes of a course given by the author at the French-Spanish School "Tresses in Pau", held in Pau (France) in October 2009. It is basically an introduction to distinct approaches and techniques that can be used to show results in braid groups. Using these techniques we provide several proofs of well known results in braid groups, namely the correctness of Artin's presentation, that the braid group is torsion free, or that its center is generated by the full twist. We also recall some solutions of the word and conjugacy problems, and that roots of a braid are always conjugate. …

来自阿贝尔几何的有限步可解群的无中心性

Center-freeness of finite-step solvable groups arising from anabelian geometry

研究双曲曲线的最大m步可解商的无中心性，证明其无挠且无中心，进而推导出m步可解Grothendieck猜想的刚性性质。