L3

  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.

 

I'll include a rather short proof of this connectedness in a group of finite

Morley rank, but I'll maybe spend most of the time talking about related matter

without giving proofs.

 

  I will discuss Garside's representation of elements of the braid group in terms of "half- twists" and the corresponding solution to the Conjugacy Problem, originally posed by Artin. If time permits, I will discuss some geometric implications of this result.