The automorphism groups of Artin groups of edge-separated CLTTF graphs

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. …

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。

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.

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.

外自同构保长的Artin群与Coxeter群的$R_\infty$性质

Property $R_\infty$ for Artin and Coxeter groups whose outer automorphisms are length-preserving

该研究证明非球面Artin与Coxeter群的保长自同构具无限Reidemeister数，进而确立多类Artin、Coxeter群的$R_\infty$性质，还证明无限不可约非仿射强刚性Coxeter群的自同构群具acylindrically双曲性。

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。

无无穷大项的大类型Artin群的自同构

Automorphisms of large-type free-of-infinity Artin groups

本文通过纯代数方式重构相关的Deligne复形，显式计算了所有无无穷大项的大类型Artin群的自同构群和外自同构群。

Automorphism groups of some affine and finite type Artin groups

We observe that, for each positive integer n > 2, each of the Artin groups of finite type A_n, B_n=C_n, and affine type \tilde A_{n-1} and \tilde C_{n-1} is a central extension of a finite index subgroup of the mapping class group of the (n+2)-punctured sphere. (The centre is trivial in the affine case and infinite cyclic in the finite type cases). Using results of Ivanov and Korkmaz on abstract commensurators of surface mapping class groups we are able to determine the automorphism groups of each member of these four infinite families of Artin groups.

右角Artin群的自同构群和外自同构群不是相对双曲的

Automorphism and outer automorphism groups of Right-angled Artin groups are not relatively hyperbolic

本文证明定义图至少含3个顶点的右角Artin群的自同构群，以及非实质上同构于单顶点或不连通定义图的外自同构群，均不是相对双曲的。

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 number of outer automorphisms of the automorphism group of a right-angled Artin group

We show that there is no uniform upper bound on |Out(Aut(A))| when A ranges over all right-angled Artin groups. This is in contrast with the cases where A is free or free abelian: for all n, Dyer-Formanek and Bridson-Vogtmann showed that Out(Aut(F_n)) = 1, while Hua-Reiner showed |Out(Aut(Z^n)| = |Out(GL(n,Z))| < 5. We also prove the analogous theorem for Out(Out(A)). We establish our results by giving explicit examples; one useful tool is a new class of graphs called austere graphs.

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…

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。

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…

关于虚拟对偶性与直角 Artin 群自同构群的一则注记

A note on virtual duality and automorphism groups of right-angled Artin groups

本文利用 Brady–Meier 判据构造外自同构群非虚拟对偶群的 RAAG，部分回答 Vogtmann 问题，并由附录介绍计算机辅助搜索更多例子。

关于$\ell^2$-Betti数和右角Artin群(外)自同构群的代数纤维化

On the $\ell^2$-Betti numbers and algebraic fibring of the (outer) automorphism group of a right-angled Artin group

本文研究了右角Artin群的$\ell^2$-Betti数及外自同构群的代数纤维化特性，揭示了其非零条件与虚拟纤维化的关系。

Finiteness of outer automorphism groups of random right-angled Artin groups

We consider the outer automorphism group Out(A_Gamma) of the right-angled Artin group A_Gamma of a random graph Gamma on n vertices in the Erdos--Renyi model. We show that the functions (log(n)+log(log(n)))/n and 1-(log(n)+log(log(n)))/n bound the range of edge probability functions for which Out(A_Gamma) is finite: if the probability of an edge in Gamma is strictly between these functions as n grows, then asymptotically Out(A_Gamma) is almost surely finite, and if the edge probability is strictly outside of both of these functions, then asymptotically Out(A_Gamma) is almost surely infinite. T…

Palindromic automorphisms of right-angled Artin groups

We introduce the palindromic automorphism group and the palindromic Torelli group of a right-angled Artin group A_G. The palindromic automorphism group Pi A_G is related to the principal congruence subgroups of GL(n,Z) and to the hyperelliptic mapping class group of an oriented surface, and sits inside the centraliser of a certain hyperelliptic involution in Aut(A_G). We obtain finite generating sets for Pi A_G and for this centraliser, and determine precisely when these two groups coincide. We also find generators for the palindromic Torelli group.

Random groups arising as graph products

In this paper we study the hyperbolicity properties of a class of random groups arising as graph products associated to random graphs. Recall, that the construction of a graph product is a generalization of the constructions of right-angled Artin and Coxeter groups. We adopt the Erdos - Renyi model of a random graph and find precise threshold functions for the hyperbolicity (or relative hyperbolicity). We aslo study automorphism groups of right-angled Artin groups associated to random graphs. We show that with probability tending to one as $n\to \infty$, random right-angled Artin groups have f…

Outer space for untwisted automorphisms of right-angled Artin groups

For a right-angled Artin group $A_Γ$, the untwisted outer automorphism group $U(A_Γ)$ is the subgroup of $Out(A_Γ)$ generated by all of the Laurence-Servatius generators except twists (where a {\em twist} is an automorphisms of the form $v\mapsto vw$ with $vw=wv$). We define a space $Σ_Γ$ on which $U(A_Γ)$ acts properly and prove that $Σ_Γ$ is contractible, providing a geometric model for $U(A_Γ)$ and its subgroups. We also propose a geometric model for all of $Out(A_Γ)$ defined by allowing more general markings and metrics on points of $Σ_Γ$.

直角Artin群的外自同构群的非柱形双曲性

Acylindrical hyperbolicity of outer automorphism groups of right-angled Artin groups

本文研究了直角Artin群外自同构群的非柱形双曲性，分别在定义图无SIL-pair和有极大SIL-pair系统时，给出了其非柱形双曲性的充要条件及分类定理，揭示了其与部分共轭存在性的关系。

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…

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。

Relative automorphism groups of right-angled Artin groups

We study the outer automorphism group of a right-angled Artin group $A_Γ$ with finite defining graph $Γ$. We construct a subnormal series for $Out(A_Γ)$ such that each consecutive quotient is either finite, free-abelian, $GL(n,\mathbb{Z})$, or a Fouxe-Rabinovitch group. The last two types act respectively on a symmetric space or a deformation space of trees, so that there is a geometric way of studying each piece. As a consequence we prove that the group $Out(A_Γ)$ is type VF (it has a finite index subgroup with a finite classifying space). The main technical work is a study of relative outer …

Arithmetic Quotients of the Automorphism Group of a Right-Angled Artin Group

It was previously shown by Grunewald and Lubotzky that the automorphism group of a free group, $\text{Aut}(F_n)$, has a large collection of virtual arithmetic quotients. Analogous results were proved for the mapping class group by Looijenga and by Grunewald, Larsen, Lubotzky, and Malestein. In this paper, we prove analogous results for the automorphism group of a right-angled Artin group for a large collection of defining graphs. As a corollary of our methods we produce new virtual arithmetic quotients of $\text{Aut}(F_n)$ for $n \geq 4$ where $k$th powers of all transvections act trivially fo…

右角Artin群的可提升自同构

Liftable automorphisms of right-angled Artin groups

研究右角Artin群的可提升自同构及其相关同态性质

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。

关于大型 Artin 群的扭曲共轭问题

On the twisted conjugacy problem for large-type Artin groups

本文证明了一类大型 Artin 群（其外自同构群有限且由特定生成元生成）的扭曲共轭问题可解，涵盖 XXXL 群及具有特定无扭曲层级结构的群。

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.

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 $n^{k+2}$.

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…

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。

直角 Artin 群的 McCullough–Miller 复形

The McCullough-Miller complex for right angled Artin groups

本文将 McCullough–Miller 的可缩复形构造从自由群推广到任意直角 Artin 群的纯对称自同构群，并展示其在上同调与不变量计算中的应用。

XL型阿廷群之间的同态

Homomorphisms between XL-type Artin groups

该文研究XL型阿廷群的同态性质，证明一般阿廷群兼具霍普夫性与余霍普夫性，刻画了标记≥6的一般图阿廷群的余霍普夫性及外自同构群有限性的条件，还给出无割点时自同构群的生成元集。

Artin groups of type B and D

We show that each of the Artin groups of type $B_n$ and $D_n$ can be presented as a semidirect product $F \rtimes {\cal B}_n$, where $F$ is a free group and ${\cal B}_n$ is the $n$-string braid group. We explain how these semidirect product structures arise quite naturally from fibrations, and observe that, in each case, the action of the braid group ${\cal B}_n$ on the free group $F$ is classical. We prove that, for each of the semidirect products, the group of automorphisms which leave invariant the normal subgroup $F$ is small: namely, ${\rm Out}(A(B_n),F)$ has order 2, and ${\rm Out}(A(D_n…

Diagonal complexes and the integral homology of the automorphism group of a free product

The main goal of this paper is a calculation of the integral (co)homology of the group of symmetric automorphisms of a free product. We proceed by giving a geometric interpretation of symmetric automorphisms via a moduli space of certain diagrams, which we name cactus products. To describe this moduli space a theory of diagonal complexes is introduced. This offers a generalisation of the theory of right-angled Artin groups in that each diagonal complex defines what we call a diagonal right-angled Artin group (DRAAG).

大类型Artin群的刚性及分类结果

Rigidity and classification results for large-type Artin groups

本文计算了许多大类型Artin群交集图的自同构群，并以此为应用推导出包括外自同构群、可公度性分类及拟等距、测度等价、轨道等价、格嵌入和交叉积von Neumann代数刚性等一系列分类与刚性结果。

Automorphism groups of some pure braid groups

We find finite presentations for the automorphism group of the Artin pure braid group and the automorphism group of the pure braid group associated to the full monomial group.

Automorphisms of two-dimensional RAAGs and partially symmetric automorphisms of free groups

We compute the virtual cohomological dimension (VCD) of the group of partially symmetric outer automorphisms of a free group. We use this to obtain new upper and lower bounds on the VCD of the outer automorphism group of a two-dimensional right-angled Artin group. In the case of a right-angled group with defining graph a tree, the bounds agree.

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.

Subgroups and quotient groups of automorphism groups of RAAGs

We study subgroups and quotients of outer automorphism groups of right-angled Artin groups (RAAGs). We prove that for all RAAGS, the outer automorphism group is residually finite and, for a large class of RAAGs, it satisfies the Tits alternative. We also investigate which of these automorphism groups contain non-abelian solvable subgroups.

Outer actions of $\mathrm{Out}(F_n)$ on small right-angled Artin groups

We determine the precise conditions under which $\mathrm{SOut}(F_n)$, the unique index two subgroup of $\mathrm{Out}(F_n)$, can act non-trivially via outer automorphisms on a RAAG whose defining graph has fewer than $\frac 1 2 \binom n 2 $ vertices. We also show that the outer automorphism group of a RAAG cannot act faithfully via outer automorphisms on a RAAG with a strictly smaller (in number of vertices) defining graph. Along the way we determine the minimal dimensions of non-trivial linear representations of congruence quotients of the integral special linear groups over algebraically clos…

Representations of Braid Groups and Generalisations

We define and study extensions of Artin's representation and braid monodromy representation to the case of topological and algebraical generalisations of braid groups. In particular we provide faithful representations of braid groups of oriented surfaces with boundary components as (outer) automorphisms of free groups. We give also similar representations for braid groups of non oriented surfaces with boundary components and we show a representation of braid groups of closed surfaces as outer automorphisms of free groups. Finally, we provide faithful representations of Artin-Tits groups of typ…

D型Artin群的自同态

Endomorphisms of Artin groups of type D

本文确定了 $n\ge 6$ 时 $D_n$ 型 Artin 群及其商群的自同态分类与自同构群，并利用拓扑论证证明了相关同态分类及商群的 co-Hopfian 性质。

Twisted Lawrence-Krammer representations

Lawrence-Krammer representations are an important family of linear representations of Artin-Tits groups of small type, which are known, under some assumptions on the parameters, to be faithful when the type is spherical (or more generally when they are restricted to the Artin-Tits monoid) and irreducible when the type is connected. Here, we investigate an analogue of these representations --- introduced by Digne in the spherical cases --- for every Artin-Tits monoid that appears as the submonoid of fixed points of an Artin-Tits monoid of small type under a group of graph automorphisms, and for…

Measure equivalence classification of transvection-free right-angled Artin groups

We prove that if two transvection-free right-angled Artin groups are measure equivalent, then they have isomorphic extension graphs. As a consequence, two right-angled Artin groups with finite outer automorphism groups are measure equivalent if and only if they are isomorphic. This matches the quasi-isometry classification. However, in contrast with the quasi-isometry question, we observe that no right-angled Artin group is superrigid for measure equivalence in the strongest possible sense, for two reasons. First, a right-angled Artin group $G$ is always measure equivalent to any graph product…

虚拟直角Artin群中的共轭问题

Conjugacy problem in virtual right-angled Artin groups

本文运用代数与几何方法，证明保长自同构构造的virtual RAAGs是CAT(0)群且共轭问题可解，推导其共轭增长级数的超越性，研究非保长情形实例，并解决保长自同构下RAAGs的扭曲共轭问题及部分算法复杂度。

A note on nilpotent subgroups of automorphism groups of RAAGs

We observe that automorphism groups of right-angled Artin groups contain nilpotent non-abelian subgroups, namely $H_3(\mathbb{Z})$ the three-dimensional integer Heisenberg group, provided they admit a certain type of element, called an adjacent transvection. This represents a (minor) extension of a result of Charney-Vogtmann.

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.

Representations of the braid group by automorphisms of groups, invariants of links, and Garside groups

From a group $H$ and a non-trivial element $h$ of $H$, we define a representation $ρ: B_n \to \Aut(G)$, where $B_n$ denotes the braid group on $n$ strands, and $G$ denotes the free product of $n$ copies of $H$. Such a representation shall be called the Artin type representation associated to the pair $(H,h)$. The goal of the present paper is to study different aspects of these representations. Firstly, we associate to each braid $β$ a group $Γ_{(H,h)} (β)$ and prove that the operator $Γ_{(H,h)}$ determines a group invariant of oriented links. We then give a topological construction of the Arti…

Dynamical systems and operator algebras associated to Artin's representation of braid groups

Artin's representation is an injective homomorphism from the braid group $B_n$ on $n$ strands into $\operatorname{Aut}\mathbb{F}_n$, the automorphism group of the free group $\mathbb{F}_n$ on $n$ generators. The representation induces maps $B_n\to\operatorname{Aut}C^*_r(\mathbb{F}_n)$ and $B_n\to\operatorname{Aut}C^*(\mathbb{F}_n)$ into the automorphism groups of the corresponding group $C^*$-algebras of $\mathbb{F}_n$. These maps also have natural restrictions to the pure braid group $P_n$. In this paper, we consider twisted versions of the actions by cocycles with values in the circle, and d…

Representing braids by automorphisms

Based on a normal form for braid group elements suggested by Dehornoy, we prove several representations of braid groups by automorphisms of a free group to be faithful. This includes a simple proof of the standard Artin's representation being faithful.

Compressed conjugacy and the word problem for outer automorphism groups of graph groups

It is shown that for graph groups (right-angled Artin groups) the conjugacy problem as well as a restricted version of the simultaneous conjugacy problem can be solved in polynomial time even if input words are represented in a compressed form. As a consequence it follows that the word problem for the outer automorphism group of a graph group can be solved in polynomial time.

性质 $R_\infty$ 对新类别的Artin群的性质

Property $R_\infty$ for new classes of Artin groups

本文研究了新类别的Artin群的$R_\infty$性质，通过自同构群和超曲面作用的分析，并提供了Delzant引理的证明。