Algebra/Topology seminar

Maria Cumplido (Rennes), Curves in surfaces and parabolic subgroups of Artin-Tits groups

Abstract: Artin-Tits groups are a natural generalisation of braid groups from the algebraic point of view: In the same way that the braid group can be obtained from the presentation of the symmetric group with transpositions as generators by dropping the order relations for the generators, other Coxeter groups give rise to more general Artin-Tits groups. If the underlying Coxeter group is finite, the resulting Artin-Tits group is said to be of spherical type. Artin-Tits groups of spherical type share many properties with braid groups.

However, some properties of braid groups are proved using topological or geometrical techniques, since a braid group can be seen as the fundamental group of a configuration space, and also as a mapping class group of a punctured disc. As one cannot replicate these topological or geometrical techniques in other Artin-Tits groups, they must be replaced by algebraic arguments, if one tries to extend properties of braid groups to all Artin-Tits groups of spherical type. That is why we are interested in parabolic subgroups of Artin-Tits groups, which are the analogue of isotopy classes of simple closed curves in the punctured disk (the building blocks that form the well-known complex of curves).  The properties of the complex of curves, and the way in which the braid group acts on it, allow to use geometric arguments to prove results in braid groups. Then, it is logical to believe that improving our understanding about parabolic subgroups will allow us to prove similar results for Artin-Tits groups of spherical type in general.

The aim of this talk is to explain the analogy mentioned before. In addition, two new results about parabolic subgroups will be presented, namely that the intersection of parabolic subgroups is a parabolic subgroup and that the set of parabolic subgroups is a lattice. This is a joint work with Volker Gebhardt, Juan González-Meneses and Bert Wiest.