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.
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.
The Tits Conjecture, proved by Crisp and Paris, states that squares of the standard generators of any Artin group generate an obvious right-angled Artin subgroup. We consider a larger set of elements consisting of all the centers of the irreducible spherical special subgroups of the Artin group, and conjecture that sufficiently large powers of those elements generate an obvious right-angled Artin subgroup. This alleged right-angled Artin subgroup is in some sense as large as possible; its nerve is homeomorphic to the nerve of the ambient Artin group. We verify this conjecture for the class of …
We introduce the class of perturbed right-angled Artin groups. These are constructed by gluing Bieri double groups into standard right-angled Artin groups. As a first application of this construction we obtain families of CAT(0) groups containing finitely presented subgroups which are not of type $\mathrm{FP}_3$, and have exponential, or polynomial Dehn functions of prescribed degree.
We determine the factorial growth rate of the number of finite index subgroups of right-angled Artin groups as a function of the index. This turns out to depend solely on the independence number of the defining graph. We also make a conjecture for right-angled Coxeter groups and prove that it holds in a limited setting.
We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result we are able to develop a Nielsen--Thurston classification for elements in the right-angled Artin group. Our analogy spans both the algebra regarding subgroups of right-angled Artin groups and mapping class groups, as well as the geometry of the extension graph and the curve graph. O…
In this note we give the quasi-isometry classification for a class of right angled Artin groups. In particular, we obtain the first such classification for a class of Artin groups with dimension larger than 2; our families exist in every dimension.
We give a necessary and sufficient condition for a graph to have a right-angled Artin group as its braid group for braid index $\ge 5$. In order to have the necessity part, graphs are organized into small classes so that one of homological or cohomological characteristics of right-angled Artin groups can be applied. Finally we show that a given graph is planar iff the first homology of its 2-braid group is torsion-free and leave the corresponding statement for $n$-braid groups as a conjecture along with few other conjectures about graphs whose braid groups of index $\le 4$ are right-angled Art…
We show that given a collection $X=\{f_1$, \ldots , $f_m\}$ of pure mapping classes on a surface $S$, there is an explicit constant N, depending only on $X$, such that their Nth powers $\{f_1^N$, \ldots , $f_m^N\}$ generate the expected right-angled Artin subgroup of MCG($S$). Moreover, we show that these subgroups are undistorted.
In all known examples of a CAT(0) group acting on CAT(0) spaces with non-homeomorphic CAT(0) visual boundaries, the boundaries are each not path connected. In this paper, we show this does not have to be the case by providing examples of right-angled Artin groups which exhibit non-unique CAT(0) boundaries where all of the boundaries are arbitrarily connected. We also prove a combination theorem for certain amalgams of CAT(0) groups to act on spaces with non-path connected visual boundaries. We apply this theorem to some right-angled Artin groups.
We show that for a sufficiently simple surface $S$, a right-angled Artin group $A(Γ)$ embeds into $\Mod(S)$ if and only if $Γ$ embeds into the curve graph $\mC(S)$ as an induced subgraph. When $S$ is sufficiently complicated, there exists an embedding $A(Γ)\to\Mod(S)$ for some $Γ$ not contained in $\mC(S)$.
We give a group theoretic characterization of geodesics with superlinear divergence in the Cayley graph of a right-angled Artin group A(G) with connected defining graph G. We use this to determine when two points in an asymptotic cone of A(G) are separated by a cut-point. As an application, we show that if G does not decompose as the join of two subgraphs, then A(G) has an infinite-dimensional space of non-trivial quasimorphisms. By the work of Burger and Monod, this leads to a superrigidity theorem for homomorphisms from lattices into right-angled Artin groups.
We prove that every right-angled Artin group embeds into the $C^{\infty}$ diffeomorphism group of the real line. As a corollary, we show every limit group, and more generally every countable residually RAAG group, embeds into the $C^{\infty}$ diffeomorphism group of the real line.
The n-string braid group of a graph X is defined as the fundamental group of the n-point configuration space of the space X. This configuration space is a finite dimensional aspherical space. A. Abrams and R. Ghrist have conjectured that this braid group is a right angled Artin group if X is planar. We prove their conjecture if X is a tree whose nodes all lie in a single interval of X.
We study the class N of graphs, the right-angled Artin groups defined on which do not contain surface subgroups. We prove that a presumably smaller class N' is closed under amalgamating along complete subgraphs, and also under adding bisimplicial edges. It follows that chordal graphs and chordal bipartite graphs belong to N'.
We prove that, aside from the obvious exceptions, the mapping class group of a compact orientable surface is not abstractly commensurable with any right-angled Artin group. Our argument applies to various subgroups of the mapping class group---the subgroups generated by powers of Dehn twists and the terms of the Johnson filtration---and additionally to the outer automorphism group of a free group and to certain linear groups.
We construct quasi-isometric embeddings from right-angled Artin groups into the outer automorphism group of a free group. These homomorphisms are in analogy with those constructed in \cite{CLM}, where the target group is the mapping class group of a surface. Toward this goal, we develop tools in the free group setting that mirror those for surface groups as well as discuss various analogs of subsurface projection; these may be of independent interest.
We give a short proof of the following theorem of Sang-hyun Kim: if $A(Γ)$ is a right-angled Artin group with defining graph $Γ$, then $A(Γ)$ contains a hyperbolic surface subgroup if $Γ$ contains an induced subgraph $\bar{C}_n$ for some $n \geq 5$, where $\bar{C}_n$ denotes the complement graph of an $n$-cycle. Furthermore, we give a new proof of Kim's co-contraction theorem.
We prove that the conjugacy problem in right-angled Artin groups (RAAGs), as well as in a large and natural class of subgroups of RAAGs, can be solved in linear-time. This class of subgroups contains, for instance, all graph braid groups (i.e. fundamental groups of configuration spaces of points in graphs), many hyperbolic groups, and it coincides with the class of fundamental groups of ``special cube complexes'' studied independently by Haglund and Wise.
We prove that finitely generated purely loxodromic subgroups of a right-angled Artin group $A(Γ)$ fulfill equivalent conditions that parallel characterizations of convex cocompactness in mapping class groups $\text{Mod}(S)$. In particular, such subgroups are quasiconvex in $A(Γ)$. In addition, we identify a milder condition for a finitely generated subgroup of $A(Γ)$ that guarantees it is free, undistorted, and retains finite generation when intersected with $A(Λ)$ for subgraphs $Λ$ of $Γ$. These results have applications to both the study of convex cocompactness in $\text{Mod}(S)$ and the way…
Consider the mapping class group $\Mod_{g,p}$ of a surface $Σ_{g,p}$ of genus $g$ with $p$ punctures, and a finite collection $\{f_1,...,f_k\}$ of mapping classes, each of which is either a Dehn twist about a simple closed curve or a pseudo-Anosov homeomorphism supported on a connected subsurface. In this paper we prove that for all sufficiently large $N$, the mapping classes $\{f_1^N,...,f_k^N\}$ generate a right-angled Artin group. The right-angled Artin group which they generate can be determined from the combinatorial topology of the mapping classes themselves. When $\{f_1,...,f_k\}$ are a…
We show that in any right-angled Artin group whose defining graph has chromatic number $k$, every non-trivial element has stable commutator length at least $1/(6k)$. Secondly, if the defining graph does not contain triangles, then every non-trivial element has stable commutator length at least $1/20$. These results are obtained via an elementary geometric argument based on earlier work of Culler.
We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter groups. More generally, we prove it for groups which are certain finite extensions or elementary amenable extensions of such groups.
We give necessary and sufficient conditions on the graph of a right-angled Artin group that determine whether the group is subgroup separable or not. Moreover, we investigate the profinite topology of the direct product of two free groups. We show that the profinite topology of the above group is strongly connected with the profinite topology of the free group of rank two.
We survey the relationship between the combinatorics and geometry of graphs and the algebraic structure of right-angled Artin groups. We concentrate on the defining graph of the right-angled Artin group and on the extension graph associated to the right-angled Artin group. Additionally, we discuss connections to geometric group theory and complexity theory. The final version of this survey will appear in "In the tradition of Thurston, vol.~II", ed.~K.~Ohshika and A.~Papadopoulos.
We consider the question of which right-angled Artin groups contain closed hyperbolic surface subgroups. It is known that a right-angled Artin group $A(K)$ has such a subgroup if its defining graph $K$ contains an $n$-hole (i.e. an induced cycle of length $n$) with $n\geq 5$. We construct another eight "forbidden" graphs and show that every graph $K$ on $\le 8$ vertices either contains one of our examples, or contains a hole of length $\ge 5$, or has the property that $A(K)$ does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a \RAAG to conta…
We survey the role of right-angled Artin groups in the theory of diffeomorphism groups of low dimensional manifolds. We first describe some of the subgroup structure of right-angled Artin groups. We then discuss the interplay between algebraic structure, compactness, and regularity for group actions on one--dimensional manifolds. For compact one--manifolds, every right-angled Artin group acts faithfully by $C^1$ diffeomorphisms, but the right-angled Artin groups which act faithfully by $C^2$ diffeomorphisms are very restricted. For the real line, every right-angled Artin group acts faithfully …
We prove that an arbitrary right-angled Artin group $G$ admits a quasi-isometric group embedding into a right-angled Artin group defined by the opposite graph of a tree. Consequently, $G$ admits quasi-isometric group embeddings into a pure braid group and into the area-preserving diffeomorphism groups of the 2--disk and the 2--sphere, answering questions due to Crisp--Wiest and M. Kapovich. Another corollary is that a pure braid group contains a closed hyperbolic manifold group as a quasi-isometrically embedded subgroup up to dimension eight. Finally, we show that the isomorphism problem, conj…
We describe sufficient conditions which guarantee that a finite set of mapping classes generate a right-angled Artin group quasi-isometrically embedded in the mapping class group. Moreover, under these conditions, the orbit map to Teichmuller space is a quasi-isometric embedding for both of the standard metrics. As a consequence, we produce infinitely many genus h surfaces (for any h at least 2) in the moduli space of genus g surfaces (for any g at least 3) for which the universal covers are quasi-isometrically embedded in the Teichmuller space.
In this paper we consider several classical and novel algorithmic problems for right-angled Artin groups, some of which are closely related to graph theoretic problems, and study their computational complexity. We study these problems with a view towards applications to cryptography.
We compute the BNS-invariant for the pure symmetric automorphism groups of right-angled Artin groups. We use this calculation to show that the pure symmetric automorphism group of a right-angled Artin group is itself not a right-angled Artin group provided that its defining graph contains a separating intersection of links.
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 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.
We construct new families of quasimorphisms on many groups acting on CAT(0) cube complexes. These quasimorphisms have a uniformly bounded defect of 12, and they "see" all elements that act hyperbolically on the cube complex. We deduce that all such elements have stable commutator length at least 1/24. The group actions for which these results apply include the standard actions of right-angled Artin groups on their associated CAT(0) cube complexes. In particular, every non-trivial element of a right-angled Artin group has stable commutator length at least 1/24. These results make use of some ne…
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…
We study the algebraic structure of the automorphism group of a general right-angled Artin group. We show that this group is virtually torsion-free and has finite virtual cohomological dimension. This generalizes results proved by the authors and John Crisp (arXiv:math/0610980) for two-dimensional 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 $Σ_Γ$.
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 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.
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.
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.
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 $T$.
We construct an embedding of any right-angled Artin group $G(Δ)$ defined by a graph $Δ$ into a graph braid group. The number of strands required for the braid group is equal to the chromatic number of $Δ$. This construction yields an example of a hyperbolic surface subgroup embedded in a two strand planar graph braid group.
Let $G$ be a free-by-cyclic group or a 2-dimensional right-angled Artin group. We provide an algebraic and a geometric characterization for when each aspherical simplicial complex with fundamental group isomorphic to $G$ has minimal volume entropy equal to 0. In the nonvanishing case, we provide a positive lower bound to the minimal volume entropy of an aspherical simplicial complex of minimal dimension for these two classes of groups. Our results rely upon a criterion for the vanishing of the minimal volume entropy for 2-dimensional groups with uniform uniform exponential growth. This criteri…
Let $Γ$ be a finite graph and let $A(Γ)$ be the corresponding right-angled Artin group. We characterize the Hamiltonicity of $Γ$ via the structure of the cohomology algebra of $A(Γ)$. In doing so, we define and develop a new canonical graph associated to a matrix, which as a consequence provides a novel perspective on the matrix determinant.
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.
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.
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.
Let G be a real semisimple Lie group with no compact factors and finite centre, and let $Λ$ be a lattice in G. Suppose that there exists a homomorphism from $Λ$ to the outer automorphism group of a right-angled Artin group $A_Γ$ with infinite image. We give an upper bound to the real rank of G that is determined by the structure of cliques in $Γ$. An essential tool is the Andreadakis-Johnson filtration of the Torelli subgroup $\mathcal{T}}(A_Γ)$ of $Aut(A_Γ)$. We answer a question of Day relating to the abelianisation of $\mathcal{T}}(A_Γ)$, and show that $\mathcal{T}}(A_Γ)$ and its image in $…
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.
