Partially commutative groups: divisibility, orthogonal systems and universal theory.

  I shall describe some joint work with Vladimir Remeslennikov and Ilia Kazachkov. Partially commutative groups are groups given by a presentation determined by a graph: vertices are generators and edges define commutation relations. Divisbility and orthogonal systems are tools developed to study these groups. Using them we have descriptions of centralisers of subsets, a good understanding of the centraliser lattice in terms of the underlying graph and have made good progress towards classifying the universal theory of these groups as well as their automorphism groups.