Abelian splittings of Right-Angled Artin Groups

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.

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.

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.

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 $A_2$.

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 $T$.

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.

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.

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.

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。

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.

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.

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。

Co-contractions of Graphs and Right-angled Artin Groups

We define an operation on finite graphs, called co-contraction. By showing that co-contraction of a graph induces an injective map between right-angled Artin groups, we exhibit a family of graphs, without any induced cycle of length at least 5, such that the right-angled Artin groups on those graphs contain hyperbolic surface groups. This gives the negative answer to a question raised by Gordon, Long and Reid.

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.

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.

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.

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.

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。

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.

直角Artin群是对称图群

Right-angled Artin groups are symmetric diagram groups

本文证明纯虚拟孪生群 $PVT_n$ 可描述为对称图群，并进一步证明每个有限生成的直角Artin群都是对称图群，揭示了其与平面图群的差异。

The $R_\infty$-property for right-angled Artin groups

Given a group $G$ and an automorphism $φ$ of $G$, two elements $x, y \in G$ are said to be $φ$-conjugate if $x = g y φ(g)^{-1}$ for some $g \in G$. The number of equivalence classes is the Reidemeister number $R(φ)$ of $φ$, and if $R(φ) = \infty$ for all automorphisms of $G$, then $G$ is said to have the $R_\infty$-property. A finite simple graph $Γ$ gives rise to the right-angled Artin group $A_Γ$, which has as generators the vertices of $Γ$ and as relations $vw = wv$ if and only if $v$ and $w$ are joined by an edge in $Γ$. We conjecture that all non-abelian right-angled Artin groups have the…

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。

On commensurability of right-angled Artin groups I: RAAGs defined by trees of diameter 4

In this paper we study the classification of right-angled Artin groups up to commensurability. We characterise the commensurability classes of RAAGs defined by trees of diameter 4. In particular, we prove a conjecture of Behrstock and Neumann that there are infinitely many commensurability classes. Hence, we give first examples of RAAGs that are quasi-isometric but not commensurable.

The lower central series of a right-angled Artin group

We give a description of Duchamp and Krob's extension of Magnus' approach to the lower central series of the free group to right-angled Artin groups. We also describe how Lalonde's extension of Lyndon words to the partially-commutative settings gives a simple algorithm to find a basis for consecutive quotients of the lower central series of a RAAG.

直角Artin群及其幂零商的$R_\infty$性质

The $R_\infty$-property for right-angled Artin groups and their nilpotent quotients

本文证明了所有非交换直角Artin群均具有$R_\infty$性质，给出了$R_\infty$-幂零指数的界，并证明了在免置换图情形下该指数等于2。

Peak reduction and finite presentations for automorphism groups of right-angled Artin groups

We generalize the peak-reduction algorithm (Whitehead's theorem) for free groups to a theorem about a general right-angled Artin group A_Gamma. As an application, we find a finite presentation for the automorphism group Aut A_Gamma that generalizes McCool's presentation for the automorphism group of a finite rank free group. We also give consider a stronger generalization of peak-reduction, giving a counterexample and proving a special case.

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.

Hereditary conjugacy separability of right angled Artin groups and its applications

We prove that finite index subgroups of right angled Artin groups are conjugacy separable. We then apply this result to establish various properties of other classes of groups. In particular, we show that any word hyperbolic Coxeter group contains a conjugacy separable subgroup of finite index and has a residually finite outer automorphism group. Another consequence of the main result is that Bestvina-Brady groups are conjugacy separable and have solvable conjugacy problem.

Right-angled Artin groups and enhanced Koszul properties

Let F be a finite field. We prove that the cohomology algebra with coefficients in F of a right-angled Artin group is a strongly Koszul algebra for every finite graph $Γ$. Moreover, the same algebra is a universally Koszul algebra if, and only if, the graph $Γ$ associated to the right-angled Artin group has the diagonal property. From this we obtain several new examples of pro-p groups, for a prime number p, whose continuous cochain cohomology algebra with coefficients in the field of p elements is strongly and universally (or strongly and non-universally) Koszul. This provides new support to …

Right Angled Artin Groups and partial commutation, old and new

We compute the $p$-central and exponent-$p$ series of all right angled Artin groups, and compute the dimensions of their subquotients. We also describe their associated Lie algebras, and relate them to the cohomology ring of the group as well as to a partially commuting polynomial ring and power series ring. We finally show how the growth series of these various objects are related to each other.

Poly-free constructions for right-angled Artin groups

We show that every right-angled Artin group AG defined by a graph G of finite chromatic number is poly-free with poly-free length bounded between the clique number and the chromatic number of G. Further, a characterization of all right-angled Artin groups of poly-free length 2 is given, namely the group AG has poly-free length 2 if and only if there exists an independent set of vertices D in G such that every cycle in G meets D at least twice. Finally, it is shown that AG is a semidirect product of 2 free groups of finite rank if and only if G is a finite tree or a finite complete bipartite gr…

松散耳分解及其在直角Artin群中的应用

Loose ear decompositions and their applications to right-angled Artin groups

本研究将图的耳分解推广为松散耳分解，用直角Artin群（RAAGs）刻画了耳分解、松散耳分解及平面图、图子式等图论概念，确定了以耳为底层图的RAAGs顶点生成元集，相关结果可通过RAAGs的松散耳分解推导得出。

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. …

Full-featured peak reduction in right-angled Artin groups

We prove a new version of the classical peak-reduction theorem for automorphisms of free groups in the setting of right-angled Artin groups. We use this peak-reduction theorem to prove two important corollaries about the action of the automorphism group of a right-angled Artin group $A_Γ$ on the set of $k$-tuples of conjugacy classes from $A_Γ$: orbit membership is decidable, and stabilizers are finitely presentable. Further, we explain procedures for checking orbit membership and building presentations of stabilizers. This improves on a previous result of the author's. We overcome a technical…

二维直角 Coxeter 群的视觉直角 Artin 子群

Visual right-angled Artin subgroups of two-dimensional right-angled Coxeter groups

本文研究二维直角 Coxeter 群中直角 Artin 有限指数子群的存在性，将其与图的卫星可拆解性联系起来，并给出构造合适子图或判定不存在的算法。

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.

Dilatation of outer automorphisms of Right-angled Artin Groups

We study the dilatation of outer automorphisms of right-angled Artin groups. Given a right-angled Artin group defined on a simplicial graph: $A(Γ) = \langle V | E \rangle$ and an automorphism $ϕ\in Out(A(Γ))$ there is a natural measure of how fast the length of a word of $A(Γ)$ grows after $n$ iterations of $ϕ$ as a function of $n$, which we call the dilatation of $w$ under $ϕ$. We define the dilatation of $ϕ$ as the supremum over dilatations of all $w \in A(Γ)$. Assuming that $ϕ$ is a pure and square map, we show that if the dilatation of $ϕ$ is positive, then either there exists a free abeli…

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…

Powers of Dehn twists generating right-angled Artin groups

We give a bound for the exponents of powers of Dehn twists to generate a right-angled Artin group. Precisely, if $\mathcal{F}$ is a finite collection of pairwise distinct simple closed curves on a finite type surface and if $N$ denotes the maximum of the intersection numbers of all pairs of curves in $\mathcal{F}$, then we prove that $\{T_γ^n \,\vert\, γ\in \mathcal{F} \}$ generates a right-angled Artin group for all $n \geq N^2 + N + 3$. This extends a previous result of Koberda, who proved the existence of a bound possibly depending on the underlying hyperbolic structure of the surface. In t…

直角 Artin 群映射环面的 Dehn 函数

Dehn functions of mapping tori of right-angled Artin groups

本文研究直角 Artin 群在自同构下的代数映射环面，并对所有 3 生成元直角 Artin 群及 F_k × F_l（k,l ≥ 2）按自同构 Φ 分类其 Dehn 函数。

Embeddings of right-angled Artin groups

We explicitly construct an embedding of a right-angled Artin group into a classical pure braid group. Using this we obtain a number of corollaries describing embeddings of arbitrary Artin groups into right-angled Artin groups and linearly independent subgroups of a right-angled Artin group.

扭化右角Artin群

Twisted right-angled Artin groups

该研究构建了扭化右角Artin群的完全重写系统，借助其导出的范式展开应用研究，对比了该类群与右角Artin群在几何、代数层面的异同。

On commensurability of some right-angled Artin groups II: RAAGs defined by paths

In this paper we continue the study of right-angled Artin groups up to commensurability initiated in [CKZ]. We show that RAAGs defined by different paths of length greater than 3 are not commensurable. We also characterise which RAAGs defined by paths are commensurable to RAAGs defined by trees of diameter 4. More precisely, we show that a RAAG defined by a path of length $n>4$ is commensurable to a RAAG defined by a tree of diameter 4 if and only if $n$ is 2 modulo 4. These results follow from the connection that we establish between the classification of RAAGs up to commensurability and line…

Right-angled Artin groups and full subgraphs of graphs

For a finite graph $Γ$, let $G(Γ)$ be the right-angled Artin group defined by the complement graph of $Γ$. We show that, for any linear forest $Λ$ and any finite graph $Γ$, $G(Λ)$ can be embedded into $G(Γ)$ if and only if $Λ$ can be realised as a full subgraph of $Γ$. We also prove that if we drop the assumption that $Λ$ is a linear forest, then the above assertion does not hold, namely, for any finite graph $Λ$, which is not a linear forest, there exists a finite graph $Γ$ such that $G(Λ)$ can be embedded into $G(Γ)$, though $Λ$ cannot be embedded into $Γ$ as a full subgraph.

Infinite groups acting faithfully on the outer automorphism group of a right-angled Artin group

We construct the first known examples of infinite subgroups of the outer automorphism group of Out(A_Gamma), for certain right-angled Artin groups A_Gamma. This is achieved by introducing a new class of graphs, called focused graphs, whose properties allow us to exhibit (infinite) projective linear groups as subgroups of Out(Out(A_Gamma)). This demonstrates a marked departure from the known behavior of Out(Out(A_Gamma)) when A_Gamma is free or free abelian, as in these cases Out(Out(A_Gamma)) has order at most 4. We also disprove a previous conjecture of the second author, producing new exampl…

Generalising some results about right-angled Artin groups to graph products of groups

We prove three results about the graph product $G=\G(Γ;G_v, v \in V(Γ))$ of groups $G_v$ over a graph $Γ$. The first result generalises a result of Servatius, Droms and Servatius, proved by them for right-angled Artin groups; we prove a necessary and sufficient condition on a finite graph $Γ$ for the kernel of the map from $G$ to the associated direct product to be free (one part of this result already follows from a result in S. Kim's Ph.D. thesis). The second result generalises a result of Hermiller and Sunic, again from right-angled Artin groups; we prove that for a graph $Γ$ with finite ch…

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…

Symplectic structures on right-angled Artin groups: between the mapping class group and the symplectic group

We define a family of groups that include the mapping class group of a genus g surface with one boundary component and the integral symplectic group Sp(2g,Z). We then prove that these groups are finitely generated. These groups, which we call mapping class groups over graphs, are indexed over labeled simplicial graphs with 2g vertices. The mapping class group over the graph Gamma is defined to be a subgroup of the automorphism group of the right-angled Artin group A_Gamma of Gamma. We also prove that the kernel of the map Aut A_Gamma to Aut H_1(A_Gamma) is finitely generated, generalizing a th…

Amenable covers of right-angled Artin groups

Let $A_L$ be the right-angled Artin group associated to a finite flag complex $L$. We show that the amenable category of $A_L$ equals the virtual cohomological dimension of the right-angled Coxeter group $W_L$. In particular, right-angled Artin groups satisfy a question of Capovilla--Löh--Moraschini proposing an inequality between the amenable category and Farber's topological complexity.

嵌入纽结群的扭曲右角阿廷群

Twisted right-angled Artin groups embedded in knot groups

本文基于混合图定义扭曲右角阿廷群，在Katayama对可嵌入纽结群的右角阿廷群刻画基础上，完全确定了可嵌入纽结群的扭曲右角阿廷群。

Commensurability invariance for abelian splittings of right-angled Artin groups, braid groups and loop braid groups

We prove that if a right-angled Artin group $A_Γ$ is abstractly commensurable to a group splitting non-trivially as an amalgam or HNN-extension over $\mathbb{Z}^n$, then $A_Γ$ must itself split non-trivially over $\mathbb{Z}^k$ for some $k\le n$. Consequently, if two right-angled Artin groups $A_Γ$ and $A_Δ$ are commensurable and $Γ$ has no separating $k$-cliques for any $k\le n$ then neither does $Δ$, so "smallest size of separating clique" is a commensurability invariant. We also discuss some implications for issues of quasi-isometry. Using similar methods we also prove that for $n\ge 4$ the…

The cohomology of right angled Artin groups with group ring coefficients

We give an explicit formula for the cohomology of a right angled Artin group with group ring coefficients in terms of the cohomology of its defining flag complex.

直角 Artin 群有理特征标的核的表示

Presentation of kernels of rational characters of right-angled Artin groups

本文刻画直角 Artin 群有理特征标核的有限生成与有限表示条件，给出有限生成集及表示，推广并代数证明了相关经典结果。

Automorphisms of two-dimensional right-angled Artin groups

We study the outer automorphism group of a right-angled Artin group A_G in the case where the defining graph G is connected and triangle-free. We give an algebraic description of Out(A_G) in terms of maximal join subgraphs in G and prove that the Tits' alternative holds for Out(A_G). We construct an analogue of outer space for Out(A_G) and prove that it is finite dimensional, contractible, and has a proper action of Out(A_G). We show that Out(A_G) has finite virtual cohomological dimension, give upper and lower bounds on this dimension and construct a spine for outer space realizing the most g…