Parabolic subgroups of large-type Artin groups

Maria Cumplido Cabello (Sevilla)

Abstract: Artin groups are a natural generalisation of braid groups from an algebraic point of view: in the same way that braids are obtained from the presentation of the symmetric group, other Coxeter groups give rise to more general Artin groups. There are very few results proven for every Artin group. To study them, specialists have focused on some special kind of subgroup, called "parabolic subgroups". These groups are used to build important simplicial complexes, as the Deligne complex or the recent complex of irreducible parabolic subgroups. The question "Is the intersection of parabolic subgroups a parabolic subgroup is a very basic question whose answer is only known for spherical Artin groups and RAAGs. In this talk, we will see how we can answer this question in Artin groups of large type, by using the geometric realisation of the poset of parabolic subgroups, that we have named "Artin complex". In particular, we will show that this complex in the large case has a property called sistolicity (a sort of weak CAT(0) property) that allows us to apply techniques from geometric group theory. This is a joint work with Alexandre Martin and Nicolas Vaskou.