Artin groups and systolic-by-function complexes

Artin groups are a generalization of braid groups and are closely related to Coxeter groups. Thanks mainly to a result of Van der Lek, a family of subgroups of Artin groups called parabolic subgroups are central to their study. One question regarding parabolic subgroups is whether the intersection of parabolic subgroups is a parabolic subgroup. This question is wide open, but the answer is known in some cases. An article by Cumplido, Martin and Vaskou introduced a geometric strategy for approaching it. In this talk we will show how to use this strategy to study it in the two-dimensional case. To do so, we will introduce systolic-by-function complexes, which are a generalization of systolic complexes, and use their non-positive curvature to get a positive answer to the question.