On parabolic subgroups of Artin groups

TL;DR

This paper investigates the intersection properties of parabolic subgroups in Artin groups, establishing new results under certain assumptions and linking these properties to fixed point phenomena and automatic continuity.

Contribution

It proves that under specific conditions, intersections of parabolic subgroups in Artin groups are parabolic, advancing understanding of their subgroup structure and related properties.

Findings

Intersections of complete and arbitrary parabolic subgroups are parabolic under certain assumptions.

Connections established between subgroup intersection behavior and fixed point properties.

Links made between subgroup structure and automatic continuity of Artin groups.

Abstract

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