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

Abstract. In the present work, 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. Also we show that right-angled Artin groups are residually torsion-free nilpotent. Moreover, we investigate the profinite topology of F2 × F2 and of the group L in [18], which are the only obstructions for the subgroup separability of the right-angled Artin groups. We show that the profinite topology of the above groups is strongly connected with the profinite topology of F2. 1.

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 Teichmüller space is a quasiisometric 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 Teichmüller space. 1

Abstract. A graph group, or right-angled Artin group, is a group given by a presentation where the only relators are commutators of the generators. A graph group presentation corresponds in a natural way to a simplicial graph, with each generator corresponding to a vertex, and each commutator relator corresponding to an edge. Suppose that G is a graph group whose corresponding graph is a tree and H is a subgroup of G. We show that if H is quasiconvex with respect to either the word metric on G or the CAT(0) metric on the universal cover of the standard complex for G, then H is separable, that is, H is the intersection of finite index subgroups of G. We also discuss some consequences relating to certain 3-manifold groups. 1.

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 ≥ 5. We construct another eight “forbidden ” graphs and show that every graph K on ≤ 8 vertices either contains one of our examples, or contains a hole of length ≥ 5, or has the property that A(K) does not contain hyperbolic closed surface subgroups. We also provide several sufficient conditions for a right angled Artin group to contain no hyperbolic surface subgroups. We prove that for one of these “forbidden ” subgraphs P2(6), the right angled Artin group A(P2(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. The first main result of the paper is a criterion for a partially commutative group G to be a domain. It allows us to reduce the study of algebraic sets over G to the study of irreducible algebraic sets, and reduce the elementary theory of G (of a coordinate group over G) to the elementary theories of the direct factors of G (to the elementary theory of coordinate groups of irreducible algebraic sets). Then we establish normal forms for quantifier-free formulas over a non-abelian directly indecomposable partially commutative group H. Analogously to the case of free groups, we introduce the notion of a generalised equation and prove that the positive theory of H has quantifier elimination and that arbitrary first-order formulas lift from H to H ∗ F, where F is a free group of finite rank. As a consequence, the positive theory of an arbitrary partially commutative group is decidable. 1.

Abstract We define graph products of families of pairs of groups and study the question when two such graph products are commensurable. As an application we prove linearity of certain graph products. AMS Classification 20F65; 57M07

Abstract. We prove that if G is a group that splits as a finite graph of finitely generated free groups with cyclic edge groups, and G has no non-Euclidean Baumslag-Solitar subgroups, then G is the fundamental group of a compact nonpositively curved cube complex. In addition, if G is also wordhyperbolic (i.e., if G contains no Baumslag-Solitar subgroups of any type), we show that G is linear (in fact, is a subgroup of SLn(Z)). Contents.

We summarize some of the main results and provide context and background for some recent results by Dani Wise on quasiconvex hierarchies for groups. These notes are based on a sequence of three talks given by Wise at the Wasatch Topology Conference in Park City, UT between 12/14 and 12/16/2009.

Embeddings of graph braid and surface groups in right-angled Artin groups and braid groups