On parabolic subgroups of Artin groups

By Philip Möller, Luis Paris, Olga Varghese. Israel Journal of Mathematics, 2022

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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.

Subfields: Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/1fc71ca01fb2fdec535ff4f2d273a1ce98f7e069