Parabolic subgroups of large-type Artin groups

TL;DR

This paper demonstrates that the poset of proper parabolic subgroups in large-type Artin groups has a systolic geometric structure, leading to new insights into subgroup stability, lattice formation, and normaliser descriptions.

Contribution

It introduces a systolic geometric framework for the poset of parabolic subgroups, establishing stability under intersections, roots, and conjugacy, and unifying normaliser results.

Findings

Poset of parabolic subgroups has systolic geometry.

Parabolic subgroups form a lattice under inclusion.

Stability of parabolic subgroups under roots and conjugacy.

Abstract

We show that the geometric realisation of the poset of proper parabolic subgroups of a large-type Artin group has a systolic geometry. We use this geometry to show that the set of parabolic subgroups of a large-type Artin group is stable under arbitrary intersections and forms a lattice for the inclusion. As an application, we show that parabolic subgroups of large-type Artin groups are stable under taking roots and we completely characterise the parabolic subgroups that are conjugacy stable. We also use this geometric perspective to recover and unify results describing the normalisers of parabolic subgroups of large-type Artin groups.