K(π, 1)'s for Artin groups of finite type

Poly-freeness of large even Artin groups

We prove that any large even Artin group is poly-free and that any even Artin group based on a triangle graph is also poly-free.

Subgroup Separability of Artin Groups

We find a condition on the underlying graph of an Artin group that fully determines if it is subgroup separable. As a consequence, an Artin group is subgroup separable if and only if it can be obtained from Artin groups of ranks at most 2 via a finite sequence of free products and direct products with the infinite cyclic group. This result generalizes the Metaftsis-Raptis criterion for Right-Angled Artin groups.

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。

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。

Abelian splittings of Right-Angled Artin Groups

We characterize when (and how) a Right-Angled Artin group splits nontrivially over an abelian subgroup.

Convexity of parabolic subgroups in Artin groups

We prove that any standard parabolic subgroup of any Artin group is convex with respect to the standard generating set.

Some virtually poly-free Artin groups

In this short note we prove that a class of Artin groups of affine and complex types are virtually poly-free, answering partially the question if all Artin groups are virtually poly-free.

Poly-freeness of even Artin groups of FC type

We prove that even Artin groups of FC type are poly-free and residually finite.

Artin groups of spherical type up to isomorphism

We prove that two Artin groups of spherical type are isomorphic if and only if their defining Coxeter graphs are the same.

Hyperbolicity of the complex of parabolic subgroups of Artin-Tits groups of type $B$

We show that the complex of parabolic subgroups associated to the Artin-Tits group of type $B$ is hyperbolic

Automorphisms of higher-dimensional right-angled Artin groups

We study the algebraic structure of the automorphism group of a general right-angled Artin group. We show that this group is virtually torsion-free and has finite virtual cohomological dimension. This generalizes results proved by the authors and John Crisp (arXiv:math/0610980) for two-dimensional right-angled Artin groups.

Embeddings of right-angled Artin groups into higher dimensional Thompson groups

In this paper, we construct embeddings of right-angled Artin groups into higher dimensional Thompson groups. In particular, we embed every right-angled Artin groups into n-dimensional Thompson group, where n is the number of complementary edges in the defining graph.

Artin groups of infinite type: trivial centers and acylindical hyperbolicity

While finite type Artin groups and right-angled Artin groups are well-understood, little is known about more general Artin groups. In this paper we use the action of an infinite type Artin group $A_Γ$ on a CAT(0) cube complex to prove that $A_Γ$ has trivial center providing the graph $Γ$ is not the star of a single vertex, and is acylindrically hyperbolic providing $Γ$ is not a join.

Asymptotic dimension of Artin groups and a new upper bound for Coxeter groups

If $A_Γ$ ($W_Γ$) is the Artin (Coxeter) group with defining graph $Γ$ we denote by $Sim(Γ)$ the number of vertices of the largest clique in $Γ$. We show that $asdimA_Γ\leq Sim(Γ)$, if $Sim(Γ)=2$. We conjecture that the inequality holds for every Artin group. We prove that if for all free of infinity Artin (Coxeter) groups the conjecture holds, then it holds for all Artin (Coxeter) groups. As a corollary, we show that $asdimW_Γ\leq Sim(Γ)$ for all Coxeter groups, which is the best known upper bound for the asymptotic dimension of Coxeter Groups. As a further corollary, we show that the asymptot…

Intersection of parabolic subgroups in even Artin groups of FC-type

We show that the intersection of parabolic subgroups of an even finitely generated FC-type Artin group is again a parabolic subgroup.

On the recognition of right-angled Artin groups

There does not exist an algorithm that can determine whether or not a group presented by commutators is a right-angled Artin group.

Abelian subgroups of two-dimensional Artin groups

We classify abelian subgroups of two-dimensional Artin groups.

Splittings of triangle Artin groups

We show that a triangle Artin group $\text{Art}_{MNP}$ where $M\leq N\leq P$ splits as an amalgamated product or an HNN extension of finite rank free groups, provided that either $M>2$, or $N>3$. We also prove that all even three generator Artin groups are residually finite.

The $Σ^1$ invariant for some Artin groups of arbitrary circuit rank

We classify the Bieri-Neumann-Strebel-Renz invariant $Σ^1(G)$ for a class of Artin groups with minimal graphs of arbitrary circuit rank.

右抵消群的度量等价分类：有限外自同构类

Measure equivalence classification of right-angled Artin groups: the finite $\mathrm{Out}$ classes

研究有限外自同构类的右抵消群的度量等价分类问题。

Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups

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.

Parabolic subgroups of Artin groups of FC type

Parabolic subgroups are the building blocks of Artin groups. This paper extends previous results, known only for parabolic subgroups of finite type Artin groups, to parabolic subgroups of FC type Artin groups. We show that the class of finite type parabolic subgroups is closed under intersection. We also study an analog of the curve complex for mapping class group constructed using parabolic subgroups. We extend the construction of the complex of parabolic subgroups to FC type Artin groups. We show that this simplicial complex is, in most cases, infinite diameter and conjecture that it is $δ-$…

定义在锥上的Artin群的中心

Centers of Artin groups defined on cones

本文证明中心猜想可从锥点集上的Artin群传递到定义图为锥的Artin群，并特别适用于仅含一个锥点的Artin群。

Right-angled Artin groups with non-path-connected boundary

We place conditions on the presentation graph of a right-angled Artin group that guarantee the standard CAT(0) cube complex on which the group acts geometrically has non-path-connected boundary.

Conjugacy stability of parabolic subgroups of Artin-Tits groups of spherical type

We give a complete classification of conjugacy stable parabolic subgroups of Artin-Tits groups of spherical type. This answers a question posed by Ivan Marin and generalizes a theorem obtained by Juan González-Meneses in the specific case of Artin braid groups.

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.

Dehn functions of subgroups of right-angled Artin groups

We show that for each positive integer $k$ there exist right-angled Artin groups containing free-by-cyclic subgroups whose monodromy automorphisms grow as $n^k$. As a consequence we produce examples of right-angled Artin groups containing finitely presented subgroups whose Dehn functions grow as $n^{k+2}$.

Conjugacy p-separability of right-angled Artin groups and applications

We prove that every subnormal subgroup of p-power index in a right-angled Artin group is conjugacy p-separable. As an application, we prove that every right-angled Artin group is conjugacy separable in the class of torsion-free nilpotent groups. As another application, we prove that the outer automorphism group of a right-angled Artin group is virtually residually p-finite. We also prove that the Torelli group of a right-angled group is residually torsion-free nilpotent, hence residually p-finite and bi-orderable.

Some definition of the Artin exponent of finite groups

The Artin exponent induced from cyclic subgroups of finite groups was studied extensively by T.Y. Lam. A Burnside ring theoretic version of Lam's results for $p$-groups was given by the author in an earlier paper. Here we look at the Artin exponent induced from the elementary abelian subgroups of finite $p$-groups using some results of A. Dress.

A finitely presented subgroup of the automorphism group of a right-angled Artin group

Let G be a right-angled Artin group. We use geometric methods to compute a presentation of the subgroup H of Aut(G) consisting of the automorphisms that send each generator to a conjugate of itself. This generalizes a result of McCool on basis-conjugating automorphisms of free groups.

Artin群的子群可分性 II

Subgroup Separability of Artin Groups II

本研究建立了判定Artin群子群可分性（LERF）的新准则，证明其等价于不含特定诱导子图，推广了Metaftsis-Raptis准则，并得出Artin群为ERF的充要条件及子群可分性与凝聚性的关联。

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 …

直角Artin群的有限商

Finite Quotients of Right-angled Artin Groups

本文证明直角Artin群在Artin群类中具有profinite刚性，即若其profinite完备化同构于某Artin群的，则该Artin群必为直角且定义图同构，方法结合pro-p完备化与有限Coxeter商。

When does a right-angled Artin group split over $\mathbb{Z}$?

We show that a right-angled Artin group, defined by a graph $Γ$ that has at least three vertices, does not split over an infinite cyclic subgroup if and only if $Γ$ is biconnected. Further, we compute JSJ--decompositions of 1--ended right-angled Artin groups over infinite cyclic subgroups.

关于虚拟Artin群的可分解性

On Decomposability of Virtual Artin Groups

本文研究了虚拟Artin群的可分解性，证明了任何连通Coxeter图对应的虚拟Artin群都是不可分解的，虚拟辫群也不例外。

Embeddability of right-angled Artin groups on the complements of linear forests

In this article, we prove that embeddings of right-angled Artin group $A_1$ on the complement of a linear forest into another right-angled Artin group $A_2$ can be reduced to full embeddings of the defining graph of $A_1$ into the extension graph of the defining graph of $A_2$.

Large-type Artin groups are systolic

We prove that Artin groups from a class containing all large-type Artin groups are systolic. This provides a concise yet precise description of their geometry. Immediate consequences are new results concerning large-type Artin groups: biautomaticity; existence of $EZ$-boundaries; the Novikov conjecture; descriptions of finitely presented subgroups, of virtually solvable subgroups, and of centralizers for infinite order elements; the Burghelea conjecture; existence of low-dimensional models for classifying spaces for some families of subgroups.

非柱形可视分裂与阿廷群的Tits二择性

Acylindrical Visual Splittings and The Tits Alternative For Artin Groups

本研究给出满足两个著名猜想条件的阿廷群可视分裂为非柱形的充要条件，借此构造出一大类满足Tits二择性的全新阿廷群实例，推进相关群论研究。

Splittings of right-angled Artin groups

We show that if a right-angled Artin group $A(Γ)$ has a non-trivial, minimal action on a tree $T$ which is not a line, then $Γ$ contains a separating subgraph $Λ$ such that $A(Λ)$ stabilizes an edge in $T$.

Isomorphisms Between Bestsvina--Brady Groups and Right-angled Artin Groups

We give a combinatorial criterion for determining which Bestvina--Brady group is isomorphic to a right-angled Artin group.

Interval Garside Structures Related to the Affine Artin Groups of Type $\tilde{A}$

Garside theory emerged from the study of Artin groups and their generalizations. Finite-type Artin groups admit two types of interval Garside structures corresponding to their standard and dual presentations. Concerning affine Artin groups, Digne established interval Garside structures for two families of these groups by using their dual presentations. Recently, McCammond established that none of the remaining dual presentations (except for one additional case) correspond to interval Garside structures. In this paper, shifting attention from dual presentations to other nice presentations for t…

Embedding Right-Angled Artin Groups into Brin-Thompson Groups

We prove that every finitely-generated right-angled Artin group can be embedded into some Brin-Thompson group $nV$. It follows that many other groups can be embedded into some $nV$ (e.g., any finite extension of any of Haglund and Wise's special groups), and that various decision problems involving subgroups of $nV$ are unsolvable.

Length functions of 2-dimensional right-angled Artin groups

Morgan and Culler proved that a minimal action of a free group on a tree is determined by its translation length function. We prove an analogue of this theorem for 2-dimensional right-angled Artin groups acting on CAT(0) rectangle complexes.

关于一类 Artin 群的等周函数

On the Isoperimetric functions of a class of Artin Groups

本文针对定义图中不含标号为3的边的Artin群，利用相对扩展表示的small cancellation理论，构造出n^6多项式等周函数，从而证明偶Artin群的字问题可解。

Cocompactly cubulated 2-dimensional Artin groups

We give a necessary and sufficient condition for a 2-dimensional or a three-generator Artin group $A$ to be (virtually) cocompactly cubulated, in terms of the defining graph of $A$.

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…

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…

XXL型Artin群的对偶Artin同构

The dual Artin isomorphism for Artin groups of XXL type

本文证明XXL型Artin群（所有定义整数m_{ij}≥5）与任意Coxeter元素对应的对偶Artin群同构，通过Hurwitz词集Q的几何性质及Coxeter群词问题的解完成证明。

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。

Subspace arrangements, BNS invariants, and pure symmetric outer automorphisms of right-angled Artin groups

We introduce a homology theory for subspace arrangements, and use it to extract a new system of numerical invariants from the Bieri-Neumann-Strebel invariant of a group. We use these to characterize when the set of basis conjugating outer automorphisms (a.k.a. the pure symmetric outer automorphism group) of a right-angled Artin group is itself a right-angled Artin group.

On the arboreal structure of right-angled Artin groups

The present article continues the study of median groups initiated in [6, 9, 10]. Some classes of median groups are introduced and investigated with a stress upon the class of the so called A-groups which contains as remarkable subclasses the lattice ordered groups and the right-angled Artin groups. Some general results concerning A-groups are applied to a systematic study of the arboreal structure of right-angled Artin groups. Structure theorems for foldings, directions, quasidirections and centralizers are proved.

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…

Conjugacy problem for subgroups with applications to Artin groups and braid type groups

Let $G$ be a group endowed with a solution to the conjugacy problem and with an algorithm which computes the centralizer in $G$ of any element of $G$. Let $H$ be a subgroup of $G$. We give some conditions on $H$, under which we provide a solution to the conjugacy problem in $H$. We apply our results to some Artin groups and braid type groups. In particular, we give explicit solutions to the conjugacy problem in the Artin groups of type $\tilde{A}_n$ and $\tilde{C}_n$.

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 …