Abelian subgroups of two-dimensional Artin groups

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…

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 parabolic subgroups of Artin-Tits groups

Abstract. We address the conjecture which states that an intersection of parabolic subgroups of an Artin-Tits group is a parabolic subgroup. We prove that the conjecture is equivalent to a, a priori, weaker conjecture. We also prove the conjecture in a specific case. Along the way, we provide short and almost self-contain algebraical proofs of several classical results on Artin-Tits groups, such as those of Van der Lek on intersection of standard parabolic subgroups.

关于Artin群的子群：偶FC情形下的导子群与柱形双曲性

Around subgroups of Artin groups: derived subgroups and acylindrical hyperbolicity in the even FC-case

该研究将直角Artin群的多项相关结果推广至更广泛的Artin群类，涵盖导子群自由性与群凝聚性的等价性、有限生成正规子群性质及FC型偶Artin群子群的柱形双曲性三方面内容。

Parabolic subgroups inside parabolic subgroups of Artin groups

We prove that a parabolic subgroup $P$ contained in another parabolic subgroup $P'$ of an Artin group $A$ is a parabolic subgroup of $P'$. This answers a question of Godelle which is not obvious despite appearances. In order to achieve our result we construct a set-retraction $A \to P$ of the inclusion map from a parabolic subgroup $P$ into $A$. This retraction was implicitly constructed in a previous paper by Charney and the second author.

Abelian splittings of Right-Angled Artin Groups

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

抛物面子群与虚拟Artin群的字问题

Parabolic subgroups and word problem in virtual Artin groups

本文研究了虚拟Artin群中标准抛物面子群的字问题，证明了在特定条件下虚拟Artin群具有可解的字问题，并探讨了强成员问题的可解性。

关于允许收缩到抛物子群的Artin群

On Artin groups admitting retractions to parabolic subgroups

本文将偶Artin群的收缩推广至FC型及更广族，证明其唯一扩展至任意抛物子群，并据此推广相交问题的简化条件，刻画FC情形的凝聚性。

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.

On solvable subgroups of automorphism groups of right-angled Artin groups

For any right-angled Artin group, we show that its outer automorphism group contains either a finite-index nilpotent subgroup or a nonabelian free subgroup. This is a weak Tits alternative theorem. We find a criterion on the defining graph that determines which case holds. We also consider some examples of solvable subgroups, including one that is not virtually nilpotent and is embedded in a non-obvious way.

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。

具有到其抛物子群的重映射的Artin群的分类

Classification of Artin groups admitting retractions onto their parabolic subgroups

本文通过分析三角子群，分类了允许到所有抛物子群重映射的Artin群，并证明此类群也允许普通重映射。

Two-generator subgroups of right-angled Artin groups are quasi-isometrically embedded

We show that non-abelian two-generator subgroups of right-angled Artin groups are quasi-isometrically embedded free groups. This provides an alternate proof of a theorem of A. Baudisch: that all two-generator subgroups are free or free abelian. Additionally, it shows that they are quasi-isometrically embedded. Our theorem also gives a method for detecting groups that are not isomorphic to a subgroup of any RAAG. We present some counterexamples in subgroups with more than two generators.

小Artin群与Coxeter群的同余子群

Congruence subgroups of small Artin and Coxeter groups

本文研究小Artin群与Coxeter群的同余子群，基于Squier矩阵表示在特定取值下的整性，推广辫群同余子群理论，并确定低层级主同余子群。

On parabolic subgroups of Artin groups

Given an Artin group $A_Γ$, a common strategy in the study of $A_Γ$ is the reduction to parabolic subgroups whose defining graphs have small diameter, i.e. showing that $A_Γ$ has a specific property if and only if all "small" parabolic subgroups of $A_Γ$ have this property. Since "small" parabolic subgroups are the puzzle pieces of $A_Γ$ one needs to study their behavior, in particular their intersections. The conjecture we address here says that the class of parabolic subgroups of $A_Γ$ is closed under intersection. Under the assumption that intersections of parabolic subgroups in complete Ar…

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 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.

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…

Parabolic subgroups of two-dimensional Artin groups and systolic-by-function complexes

We extend previous results by Cumplido, Martin and Vaskou on parabolic subgroups of large-type Artin groups to a broader family of two-dimensional Artin groups. In particular, we prove that an arbitrary intersection of parabolic subgroups of a $(2,2)$-free two-dimensional Artin group is itself a parabolic subgroup. An Artin group is $(2,2)$-free if its defining graph does not have two consecutive edges labeled by $2$. As a consequence of this result, we solve the conjugacy stability problem for this family by applying an algorithm introduced by Cumplido. All of this is accomplished by consider…

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.

直角阿廷群作为其外自同构群的有限指数子群

Right-angled Artin groups as finite-index subgroups of their outer automorphism groups

本文证明每个直角阿廷群都是另一个直角阿廷群的外自同构群的有限指数子群，并利用纯对称外自同构群给出显式构造。

关于一个关系群中右角艺术子群的注记

A note on right-angled Artin subgroups of one-relator groups

本文通过基本的Bass-Serre理论和一个关系群的经典性质，证明了当右角艺术群嵌入到一个一个关系群中时，Γ是一个有限森林。

The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group

It was conjectured by Tits that the only relations amongst the squares of the standard generators of an Artin group are the obvious ones, namely that a^2 and b^2 commute if ab=ba appears as one of the Artin relations. In this paper we prove Tits' conjecture for all Artin groups. More generally, we show that, given a number m(s)>1 for each Artin generator s, the only relations amongst the powers s^m(s) of the generators are that a^m(a) and b^m(b) commute if ab=ba appears amongst the Artin relations.

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。

阿廷群抛物子群研究中的基本技术

Fundamental techniques in the study of parabolic subgroups of Artin groups

介绍在阿廷群抛物子群研究中有效的技术，着重于抛物子群交集问题，突出核心思想与策略，为读者提供研究基本方法的切入点。

具有阿贝尔商的直角阿廷群的有限生成核

Finitely presented kernels of right-angled Artin groups with abelian quotients

研究具有阿贝尔商的直角阿廷群的核何时有限生成及有限表示，给出明确生成集与表示，还推导出其特征标核的有限表示，推广了相关群的表示。

Limit groups over coherent right-angled Artin groups are cyclic subgroup separable

We prove that cyclic subgroup separability is preserved under exponential completion for groups that belong to a class that includes all coherent RAAGs and toral relatively hyperbolic groups; we do so by exploiting the structure of these completions as iterated free products with commuting subgroups. From this we deduce that the cyclic subgroups of limit groups over coherent RAAGs are separable, answering a question of Casals-Ruiz, Duncan and Kazachov. We also discuss relations between free products with commuting subgroups and the word problem, and recover the fact that limit groups over cohe…

Strong quasiconvexity, stability, and lower relative divergence in right-angled Artin groups

Let $Γ$ be a simplicial, finite, connected graph such that $Γ$ does not decompose as a nontrivial join. We prove that two notions of strong quasiconvexity and stability are equivalent in the right-angled Artin group $A_Γ$ (except for the case of finite index subgroups). We also characterize non-trivial strongly quasiconvex subgroups of infinite index in $A_Γ$ (i.e. non-trivial stable subgroups in $A_Γ$) by quadratic lower relative divergence. These results strengthen the work of Koberda-Mangahas-Taylor on characterizing purely loxodromic subgroups of right-angled Artin groups.

Undecidability of the word problem for one-relator inverse monoids via right-angled Artin subgroups of one-relator groups

We prove the following results: (1) There is a one-relator inverse monoid $\mathrm{Inv}\langle A\:|\:w=1 \rangle$ with undecidable word problem; and (2) There are one-relator groups with undecidable submonoid membership problem. The first of these results answers a problem originally posed by Margolis, Meakin and Stephen in 1987.

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 …

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。

Local indicability and commutator subgroups of Artin groups

We determine which of the finite-type Artin groups are locally indicable, and compute presentations for their commutator subgroups.

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.

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.

Bounded rank subgroups of Coxeter groups, Artin groups and one-relator groups with torsion

We obtain a number of results regarding freeness, quasiconvexity and separability for subgroups of Coxeter groups, Artin groups and one-relator groups with torsion.

Parabolic subgroups of Garside groups

A Garside monoid is a cancellative monoid with a finite lattice generating set; a Garside group is the group of fractions of a Garside monoid. The family of Garside groups contains the Artin-Tits groups of spherical type. We generalise the well-known notion of a parabolic subgroup of an Artin-Tits group into that of a parabolic subgroup of a Garside group. We also define the more general notion of a Garside subgroup of a Garside group, which is related to the notion of LCM-homomorphisms between Artin-Tits groups. We prove that most of the properties of parabolic subgroups extend to this subgro…

Homology of subgroups of right-angled Artin groups

We describe the (co)homology of a certain family of normal subgroups of right-angled Artin groups that contain the commutator subgroup, as modules over the quotient group. We do so in terms of (skew) commutative algebra of squarefree monomial ideals.

On finitely generated normal subgroups of right-angled Artin groups and graph products of groups

A classical result of Schreier states that nontrivial finitely generated normal subgroups of free groups are of finite index, that is, free groups can only quotient to finite groups with finitely generated kernel. In this note we extend this result to the class of right-angled Artin groups (RAAGs). More precisely, we prove that the quotient of a RAAG by a finitely generated (full) normal subgroup is abelian-by-finite and finite-by-abelian. As Schreier's result extends to nontrivial free products of groups, we further show that our result extends to graph products of groups. As a corollary, we …

单关系群中的直角Artin子群与自由积

Right-angled Artin subgroups and free products in one-relator groups

该研究探讨单关系群包含直角Artin子群的准则，证明森林型图对应的正子幺半群可嵌入则全群可嵌入，还得到两类单关系群的刻画结果。

On strongly quasiconvex subgroups

We develop a theory of \emph{strongly quasiconvex subgroups} of an arbitrary finitely generated group. Strong quasiconvexity generalizes quasiconvexity in hyperbolic groups and is preserved under quasi-isometry. We show that strongly quasiconvex subgroups are also more reflexive of the ambient groups geometry than the stable subgroups defined by Durham-Taylor, while still having many analogous properties to those of quasiconvex subgroups of hyperbolic groups. We characterize strongly quasiconvex subgroups in terms of the lower relative divergence of ambient groups with respect to them. We also…

On subgroups of right angled Artin groups with few generators

For each natural number $d$ we construct a $3$-generated group $H_d$, which is a subdirect product of free groups, such that the cohomological dimension of $H_d$ is $d$. Given a group $F$ and a normal subgroup $N \lhd F$ we prove that any right angled Artin group containing the special HNN-extension of $F$ with respect to $N$ must also contain $F/N$. We apply this to construct, for every $d \in \mathbb{N}$, a $4$-generated group $G_d$, embeddable into a right angled Artin group, such that the cohomological dimension of $G_d$ is $2$ but the cohomological dimension of any right angled Artin grou…

FC型偶Artin群的子群

Subgroups of even Artin groups of FC-type

本文证明了FC型偶Artin群的子群满足Tits二择一定理，并证明其抛物子群在取根下封闭。

On the commutator subgroup of a right-angled Artin group

We use polyhedral product models to analyse the structure of the commutator subgroup of a right-angled Artin group. In particular, we provide a minimal set of generators for the commutator subgroup, consisting of special iterated commutators of canonical generators.

On embeddings between outer automorphism groups of right-angled Artin groups

Using abelian coverings of Salvetti complexes, embeddings of outer automorphism groups of right-angled Artin groups (RAAGs) into outer automorphism groups of their particular characteristic subgroups are constructed. Virtual embeddings of outer automorphism groups of finitely generated groups having the unique root property into outer automorphism groups of their particular subgroups are also given. These results provide us with rich examples of (virtual) embeddings between outer automorphism groups of RAAGs. Finite subgroups of pure (outer) automorphism groups of RAAGs are also investigated. …

Dehn functions and finiteness properties of subgroups of perturbed right-angled Artin groups

We introduce the class of perturbed right-angled Artin groups. These are constructed by gluing Bieri double groups into standard right-angled Artin groups. As a first application of this construction we obtain families of CAT(0) groups containing finitely presented subgroups which are not of type $\mathrm{FP}_3$, and have exponential, or polynomial Dehn functions of prescribed degree.

Subgroup growth of right-angled Artin and Coxeter groups

We determine the factorial growth rate of the number of finite index subgroups of right-angled Artin groups as a function of the index. This turns out to depend solely on the independence number of the defining graph. We also make a conjecture for right-angled Coxeter groups and prove that it holds in a limited setting.

Geometric embedding properties of Bestvina-Brady subgroups

We compute the relative divergence and the subgroup distortion of Bestvina-Brady subgroups. We also show that for each integer $n\geq 3$, there is a free subgroup of rank $n$ of some right-angled Artin group whose inclusion is not a quasi-isometric embedding. This result answers the question of Carr about the minimum rank $n$ such that some right-angled Artin group has a free subgroup of rank $n$ whose inclusion is not a quasi-isometric embedding. It is well-known that a right-angled Artin group $A_Γ$ is the fundamental group of a graph manifold whenever the defining graph $Γ$ is a tree. We sh…

Polyhedral products and commutator subgroups of right-angled Artin and Coxeter groups

We construct and study polyhedral product models for classifying spaces of right-angled Artin and Coxeter groups, general graph product groups and their commutator subgroups. By way of application, we give a criterion of freeness for the commutator subgroup of a graph product group, and provide an explicit minimal set of generators for the commutator subgroup of a right-angled Coxeter group.

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. …

直角Coxeter群和直角Artin群的直角Artin子群

Right-angled Artin subgroups of right-angled Coxeter and Artin groups

本文研究了直角Coxeter群和直角Artin群的子群何时为RAAGs，刻画了2维RACGs的有限指数视觉RAAG子群，并应用此结果证明了2维单端RACG拟等距于RAAG的充要条件，同时给出了非平面定义图的新例子及Dyer结果的新证明。

二维Artin群的Tits备选性质与Wise的幂备选性质

The Tits Alternative for two-dimensional Artin groups and Wise's Power Alternative

本文证明二维Artin群满足加强的Tits备选性质，并在相关Coxeter群双曲时，肯定回答Wise关于两个元素大幂生成子群的问题。

Metric systolicity and two-dimensional Artin groups

We introduce the notion of metrically systolic simplicial complexes. We study geometric and large-scale properties of such complexes and of groups acting on them geometrically. We show that all two-dimensional Artin groups act geometrically on metrically systolic complexes. As direct corollaries we obtain new results on two-dimensional Artin groups and all their finitely presented subgroups: we prove that the Conjugacy Problem is solvable, and that the Dehn function is quadratic. We also show several large-scale features of finitely presented subgroups of two-dimensional Artin groups, lying ba…

局部可约 Artin 群中的负曲率

Negative curvature in locally reducible Artin groups

本文通过定义2-完全 Artin 复形并证明其 systolic 性，进而证明一类抛物子群为弱 malnormal，最终得出许多局部可约 Artin 群是 acylindrically 双曲的结论。

到抛物子群的收缩及其应用

Retraction to a parabolic subgroup and applications

本文利用 Artin 群到标准抛物子群的左、右收缩，研究抛物子群的极小性、交、双陪集及共轭类等结构问题并获得新结论。