Surface Braid Groups and Mapping Class Groups

Mini course by Professor Paolo Bellingeri Université de Caen

Surface braid groups are a natural generalization of classical
braid groups and of fundamental groups of surfaces. They were first
defined by Zariski during the 1930’s (although braid groups on the
sphere had been considered much earlier by Hurwitz), and they were
re-discovered during the 1960’s in the study of mapping class groups and
configuration spaces. These groups, introduced as an “algebraic” tool,
turned out to be very difficult to understand. It is now common to use
mapping class techniques to study the properties of surface braid groups.

In the last decade the interest in these groups grew notably, in
particular due to their relations with knot theory and mapping class
groups and, quite astonishingly, with robotics. The mini course will
start with different definitions and group presentations for these
groups; this first part will allow us to present several combinatorial
properties, such as residual properties, central series, and related
quotients, which will lead to some applications to finite type invariant
theory as well as linear and “symmetric” representations for surface
braid groups. We will then discuss the relation between surface braid
groups and mapping class groups. We will end with an overview on classic
and more recent applications to knot theory in 3-manifolds. Here and
there we will present some open questions.

The lecturer of the mini course, Paolo Bellingeri, is a professor at the
University of Caen and the director of the Federation of Normandy
Laboratories of Mathematics. He also serves as the lead scientist of the
research group “Algèbre Représentations et Topologie pour l’Informatique
Quantique et Classique” (ARTIQ). He has published widely on the topic of
the mini course and is one of the organizers of the conference series
“WinterBraids” on this research area.