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.