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.
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 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 $T$.
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.
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.
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…
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。
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.
