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.
We show that the Aut-invariant word norm on right angled Artin and right angled Coxeter groups is unbounded (except in few special cases). To prove unboundedness we exhibit certain characteristic subgroups. This allows us to find unbounded quasimorphisms which are Lipschitz with respect to the Aut-invariant word norm.
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 $n^{k+2}$.
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.
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 $A_2$.
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 $T$.
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.
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.
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.
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.
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…
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.
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.
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.
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 …
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.
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. …
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.
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…
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…
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…
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.
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
证明了图$Γ$上群$G_v$的图积$G=\G(Γ;G_v, v \in V(Γ))$的三个结果.第一个结果推广了Servatius，Droms和Servatius对直角Artin群所证明的一个结果;我们证明了有限图$Γ$上从$G$到相应的直积的映射的核是自由的一个充要条件（这个结果的一部分已经从S.金博士thesis）。第二个结果推广了Hermiller和Sunic的一个结果，再次从直角Artin群;我们证明了对于一个具有有限色数的图$Γ$，$G$有一个级数，其中每个因子是顶点群的自由积。第三个结果提供了一个定理的另一种证明，由于迈耶，它提供了充要条件的有限图$Γ$为$G$是双曲的。
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…
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.
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…
