Seminario Teoría de grupos UAM-ICMAT

Jueves, 12 de Abril, Aula 420 (UAM), 15:00

Ponente: María Cumplido Cabello (Université Rennes 1 / Universidad de)

Título: Parabolic subgroups of Artin-Tits groups of spherical type

Abstract: (with Volker Gebhardt, Juan González-Meneses and Bert Wiest)
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 of these properties for the braid group are proved using topological or geometrical techniques, since a braid group can be seen as the fundamental group of a configuration 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 when trying to extend these properties to all Artin-Tits groups of spherical type. That is why we are interested in parabolic subgroups of Artin-Tits groups, which are defined as  conjugates of a subgroups generated by a subset of the standard generators. They are the analogue of isotopy classes of simple closed curves in the puncture disk, which are the building blocks that form the well-known complex of curves. 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.

In this seminar we present the new "complex of irreducible parabolic subgroups" and  two new results, namely that the intersection of parabolic subgroups is a parabolic subgroup and that the set of parabolic subgroups is a lattice.