Kai-Uwe Bux, Ruth Charney, Karen Vogtmann
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.
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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.
We study representations of the pure symmetric automorphism group $PAut(A_Γ)$ of a RAAG $A_Γ$ with defining graph $Γ$. We first construct a homomorphism from $PAut(A_Γ)$ to the direct product of a RAAG and a finite direct product of copies of $F_2 \times F_2$; moreover, the image of $PAut(A_Γ)$ under this homomorphism is surjective onto each factor. As a consequence, we obtain interesting actions of $PAut(A_Γ)$ on non-positively curved spaces We then exhibit, for connected $Γ$, a RAAG which property contains $Inn(A_Γ)$ and embeds as a normal subgroup of $PAut(A_Γ)$. We end with a discussion of…
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.
We describe an algorithm to find the virtual cohomological dimension of the automorphism group of a right-angled Artin group. The algorithm works in the relative setting; in particular it also applies to untwisted automorphism groups and basis-conjugating automorphism groups. The main new tool is the construction of free abelian subgroups of certain Fouxe-Rabinovitch groups of rank equal to their virtual cohomological dimension, generalizing a result of Meucci in the setting of free groups.
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.
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. …
Divisible cube complexes and finite-order automorphisms of RAAGs
研究了RAAGs有限阶自同构的固定子群的几何特征，发现这些子群是可除立方复形的基群，并指出表面群和右角Coxeter群的交换子群是此类固定子群的例子。
Outer automorphism groups of RAAGs, denoted $Out(A_Γ)$, interpolate between $Out(F_n)$ and $GL_n(\mathbb{Z})$. We consider several vastness properties for which $Out(F_n)$ behaves very differently from $GL_n(\mathbb{Z})$: virtually mapping onto all finite groups, SQ-universality, virtually having an infinite dimensional space of homogeneous quasimorphisms, and not being boundedly generated. We give a neccessary and sufficient condition in terms of the defining graph $Γ$ for each of these properties to hold. Notably, the condition for all four properties is the same, meaning $Out(A_Γ)$ will eit…
We construct free abelian subgroups of the group $U(A_Γ)$ of untwisted outer automorphisms of a right-angled Artin group, thus giving lower bounds on the virtual cohomological dimension. The group $U(A_Γ)$ was previously studied by Charney, Stambaugh and the second author, who constructed a contractible cube complex on which it acts properly and cocompactly, giving an upper bound for the virtual cohomological dimension. The ranks of our free abelian subgroups are equal to the dimensions of the principal cubes in this complex. These are often of maximal dimension, so that the upper and lower bo…
The JSJ decomposition encodes the automorphisms and the virtually cyclic splittings of a hyperbolic group. For general finitely presented groups, the JSJ decomposition encodes only their splittings. In this sequence of papers we study the automorphisms of a hierarchically hyperbolic group that satisfies some weak acylindricity conditions. To study these automorphisms we construct an object that can be viewed as a higher rank JSJ decomposition. In the first paper we demonstrate our construction in the case of a right angled Artin group. For studying automorphisms of a general HHG we construct w…
Pure symmetric automorphisms, extensions of RAAGs, and Koszulness
本文通过图上的组合条件刻画了RAAG的纯对称自同构群及其外自同构群具有Koszul降中心李代数的情形，证明其是RAAG的迭代扩张（多重自由），并指出$\mathrm{PAut}(F_n)$（$n\geq4$）非多重有限生成自由，且相关群类具有1-形式性。
Trees, homology, and automorphism groups of RAAGs
本文针对定义图为树的 RAAG，研究其自同构群中显式有限指数子群的同调，利用深顶点的数量与度数给出第一 Betti 数下界，并组合分析其随树规模变化的平均值。
Outer space and finiteness properties for symmetric automorphisms of RAAGs, and generalisations
本文为直角Artin群构造了对称外自同构群的可缩外空间骨架，并利用其证明置换给定有限共轭类集合的外自同构子群具有VF型有限性，同时确定了对称外自同构群的虚拟上同调维数。
We show that the outer automorphism groups of graph products of finitely generated abelian groups satisfy the Tits alternative, are residually finite, their so-called Torelli subgroups are finitely generated, and they satisfy a dichotomy between being virtually nilpotent and containing a non-abelian free subgroup that is determined by a graphical condition on the underlying labelled graph. Graph products of finitely generated abelian groups simultaneously generalize right-angled Artin groups (RAAGs) and right-angled Coxter groups (RACGs), providing a common framework for studying these groups.…
Finite groups of untwisted outer automorphisms of RAAGs
本文证明直角 Artin 群外自同构群中由 Whitehead 自同构和反演生成的子群的任何有限子群都固定可缩空间中的一点，从而推广了 Outer Space 的相应结果并推出有限子群共轭类有限。
We characterize groups quasi-isometric to a right-angled Artin group $G$ with finite outer automorphism group. In particular all such groups admit a geometric action on a $CAT(0)$ cube complex that has an equivariant "fibering" over the Davis building of $G$.
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 …
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。
For every finite graph $Γ$, we define a simplicial complex associated to the outer automorphism group of the RAAG $A_Γ$. These complexes are defined as coset complexes of parabolic subgroups of $Out^0(A_Γ)$ and interpolate between Tits buildings and free factor complexes. We show that each of these complexes is homotopy Cohen-Macaulay and in particular homotopy equivalent to a wedge of d-spheres. The dimension d can be read off from the defining graph $Γ$ and is determined by the rank of a certain Coxeter subgroup of $Out^0(A_Γ)$. In order to show this, we refine the decomposition sequence for…
Realising VCD for untwisted automorphism groups of RAAGs
本文研究了RAAGs无扭子群的虚拟上同调维数，通过等变变形退缩构造了新的合同立方复形，并给出了其维数与虚拟上同调维数关系的条件。
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…
$R_{\infty}$-property for groups commensurable to nilpotent quotients of RAAGs
本文研究有限生成无挠幂零群中 $R_{\infty}$-性质在抽象可公度下的不变性，通过构造与边加权图相关的 2 步幂零群反例，证明该性质在抽象可公度下不保持。
Coarse-median preserving automorphisms
本文引入保持粗中位数的自同构，刻画 RAAG/RACG 不动子群的几何性质，并将无限阶外自同构实现为有限秩中位数空间上的同位似。
The structure of a certain subgroup $S$ of the automorphism group of a partially commutative group (RAAG) $G$ is described in detail: namely the subgroup generated by inversions and elementary transvections. We define admissible subsets of the generators of $G$, and show that $S$ is the subgroup of automorphisms which fix all subgroups $\langle Y\rangle$ of $G$, for all admissible subsets $Y$. A decomposition of $S$ as an iterated tower of semi-direct products in given and the structure of the factors of this decomposition described. The construction allows a presentation of $S$ to be computed…
Quasi-isometry classification of right-angled Artin groups II: several infinite out cases
针对哪类右角Artin群（RAAG）的拟等距分类与可公度分类一致的问题，本文确定两类外自同构群含相邻平延和部分共轭的无限RAAG，证明其拟等距等价于可公度，提出基于定义图的判断算法，并通过形变论证和立方化技术解决重构映射问题。
For any right-angled Artin group $A_Γ$ we construct a finite-dimensional space $\mathcal{O}_Γ$ on which the group $\text{Out}(A_Γ)$ of outer automorphisms of $A_Γ$ acts with finite point stabilizers. We prove that $\mathcal{O}_Γ$ is contractible, so that the quotient is a rational classifying space for $\text{Out}(A_Γ)$. The space $\mathcal{O}_Γ$ blends features of the symmetric space of lattices in $\mathbb{R}^n$ with those of Outer space for the free group $F_n$. Points in $\mathcal{O}_Γ$ are locally CAT(0) metric spaces that are homeomorphic (but not isometric) to certain locally CAT(0) cub…
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。
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。
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。
Let $G$ be a right-angled Artin group with defining graph $Γ$ and let $H$ be a finitely generated group quasi-isometric to $G(Γ)$. We show if $G$ satisfies (1) its outer automorphism group is finite; (2) $Γ$ does not have induced 4-cycle; (3) $Γ$ is star-rigid; then $H$ is commensurable to $G$. We show condition (2) is sharp in the sense that if $Γ$ contains an induced 4-cycle, then there exists an $H$ quasi-isometric to $G(Γ)$ but not commensurable to $G(Γ)$. Moreover, one can drop condition (1) if $H$ is a uniform lattice acting on the universal cover of the Salvetti complex of $G(Γ)$. As a …
The goal of this paper is to construct and describe certain arithmetic subgroups of the automorphism group of a partially commutative group. More precisely, given an arbitrary finite graph $Γ$ we construct an arithmetic subgroup $St(L(G))$, represented as a subgroup of $GL(n,Z)$, where $n$ is the number of vertices of the graph $Γ$. In the last section of the paper we give a description of the decomposition of the group of automorphisms $St^{conj}(L(G))$ as a semidirect product of the group of conjugating automorphisms $Conj(G)$ and $St(L(G))$. This result is closely related to Theorem 1.4 of …
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.
Let F_{n+m} be the free group of rank n+m, with generators x_1,...,x_{n+m}. An automorphism ϕof F_{n+m} is called partially symmetric if for each 1 \le i \le m, ϕ(x_i) is conjugate to x_j or x_j^{-1} for some 1 \le j \le m. Let Σ\Aut_n^m be the group of partially symmetric automorphisms. We prove that for any m \ge 0 the inclusion Σ\Aut_n^m \to Σ\Aut_{n+1}^m induces an isomorphism in rational homology for dimensions i satisfying n \ge (3(i+1)+m)/2, with a similar statement for the groups PΣ\Aut_n^m of pure partially symmetric automorphisms. We also prove that for any n \ge 0 the inclusion Σ\Au…
We define several "standard" subgroups of the automorphism group Aut(G) of a partially commutative (right-angled Artin) group and use these standard subgroups to describe decompositions of Aut(G). If C is the commutation graph of G, we show how Aut(G) decomposes in terms of the connected components of C: obtaining a particularly clear decomposition theorem in the special case where C has no isolated vertices. If C has no vertices of a type we call dominated then we give a semi-direct decompostion of Aut(G) into a subgroup of locally conjugating automorphisms by the subgroup stabilising a certa…
A countable poset is ultrahomogeneous if every isomorphism between its finite subposets can be extended to an automorphism. The groups $\operatorname{Aut}(A)$ of such posets $A$ have a natural topology in which $\operatorname{Aut}(A)$ are Polish topological groups. We consider the problem whether $\operatorname{Aut}(A)$ contains a dense free subgroup of two generators. We show that if $A$ is ultrahomogeneous, then $\operatorname{Aut}(A)$ contains such subgroup. Moreover, we characterize whose countable ultrahomogeneous posets $A$ such that for each natural $m$, the set of all cyclically dense …
Given a finite graph G there is a corresponding group given by the presentation with generators the vertices of G and a relation [x,y]=1 for generators x and y precisely when (x,y) is an edge of G. Such groups are known as partially commutative groups (or right-angled Artin groups). In this paper we construct orthogonality theory for graphs with the study of partially commutative groups in mind. The theory developed here provides tools for the study of the structure of the centraliser lattice of partially commutative groups and for their automorphism groups.
Stable Extensions of Complete Groups
本文研究了通过中心因子群构造稳定群的方法，分类了有限稳定群，并证明了存在无限多个此类群。
Uniform difference between the length spectra of Out(F2) and the genus two handlebody group
本文研究了双曲面二柄体群与 Out(F2) 之间长度谱的均匀差异，揭示了伪 Anosov 映射的伸缩因子与完全不可约外自同构之间的关系。
Coherent configurations (CCs) are highly regular colorings of the set of ordered pairs of a "vertex set"; each color represents a "constituent digraph." CCs arise in the study of permutation groups, combinatorial structures such as partially balanced designs, and the analysis of algorithms; their history goes back to Schur in the 1930s. A CC is primitive (PCC) if all its constituent digraphs are connected. We address the problem of classifying PCCs with large automorphism groups. This project was started in Babai's 1981 paper in which he showed that only the trivial PCC admits more than $\exp(…
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