Treffer: On parabolic subgroups of Artin groups

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