Abstract:
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.

Abstract:
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.

Abstract:
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.

Abstract:
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.

Abstract:
We describe the (co)homology of a certain family of normal subgroups of right-angled Artin groups that contain the commutator subgroup, as modules over the quotient group. We do so in terms of (skew) commutative algebra of squarefree monomial ideals.

Abstract:
Recently, right-angled Artin groups have attracted much attention in geometric group theory. They have a rich structure of subgroups and nice algorithmic properties, and they give rise to cubical complexes with a variety of applications. This survey article is meant to introduce readers to these groups and to give an overview of the relevant literature.

Abstract:
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.

Abstract:
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 contain no hyperbolic surface subgroups. We prove that for one of these "forbidden" subgraphs $P_2(6)$, the right angled Artin group $A(P_2(6))$ is a subgroup of a (right angled Artin) diagram group. Thus we show that a diagram group can contain a non-free hyperbolic subgroup answering a question of Guba and Sapir. We also show that fundamental groups of non-orientable surfaces can be subgroups of diagram groups. Thus the first integral homology of a subgroup of a diagram group can have torsion (all homology groups of all diagram groups are free Abelian by a result of Guba and Sapir).

Abstract:
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 group, containing $G_d$, is at least $d$. These examples are used to show the non-existence of certain "universal" right angled Artin groups. We also investigate finitely presented subgroups of direct products of limit groups. In particular we show that for every $n\in \mathbb{N}$ there exists $\delta(n) \in \mathbb{N}$ such that any $n$-generated finitely presented subgroup of a direct product of finitely many free groups embeds into the $\delta(n)$-th direct power of the free group of rank $2$. As another corollary we derive that any $n$-generated finitely presented residually free group embeds into the direct product of at most $\delta(n)$ limit groups.