Braid groups of surfaces. (Nikolai Ivanov)


Braid groups of surfaces are the fundamental groups of the configuration spaces of distinct points on a surface. They are a common generalization of the Artin braid groups (corresponding to the case where the surface in question is a disc) and of the fundamental groups of surfaces (corresponding to the case when we consider configurations of 1 point). We will discuss extensions to to the braid groups of surfaces of the following classical results: Artin theorem about the permutation representations and the Dyer-Grossman theorem about automorphisms of the braid groups; the Dehn-Nielsen theorem about automorphisms of the fundamental groups of surfaces. An interesting feature arises in the case of surfaces with boundary: in contrast with the fundamental groups of surfaces, the higher braid groups "know" their peripheral element.

The automorphisms of the braid groups in the case of closed surfaces were studied in a joint work with E. Irmak and J. McCarthy; here we will focus on a somewhat different approach allowing us to deal with the non-closed case.