On parabolic subgroups of Artin groups

Given an Artin group A_Γ, a common strategy in the study of A_Γ is the reduction to parabolic subgroups whose defining graphs have small diameter, i.e., showing that A_Γ has a specific property if and only if all “small” parabolic subgroups of A_Γ have this property. Since “small” parabolic subgroups are the building blocks of A_Γ one needs to study their behavior, in particular their intersections. The conjecture we address here says that the class of parabolic subgroups of A_Γ is closed under intersection. Under the assumption that intersections of parabolic subgroups in complete Artin groups are parabolic, we show that the intersection of a complete parabolic subgroup with an arbitrary parabolic subgroup is parabolic. Further, we connect the intersection behavior of complete parabolic subgroups of A_Γ to fixed point properties and to automatic continuity of A_Γ using Bass–Serre theory and a generalization of the Deligne complex.