Parabolic subgroups of large-type Artin groups

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.

Parabolic subgroups of two‐dimensional Artin groups and systolic‐by‐function complexes

“…Remark In [10], they show the previous result for $$m_{st}\in \lbrace 3,4,\dots ,\infty \rbrace$$, since they work with large‐type Artin groups. However, the result also holds for the case $$m_{st}=2$$, and the proof is the same as in the other cases.…”

“…As an immediate consequence, the results of Theorem 1.1 hold for all (2,2)‐free two‐dimensional Artin groups (the cases with less than 3 generators were established in [9, 10]). In particular, we derive the main result of this article.…”

“…These last facts were also proven in [9] in the context of spherical Artin groups. The proofs presented in [10] are a geometrical reinterpretation of the latter. The second item had also been proven previously by Godelle in [12].…”

“…In order to show that large‐type Artin groups satisfy the conditions of the theorem, Cumplido, Martin and Vaskou proved that Artin complexes are systolic in the sense of [15], and used the fact that if a group $$G$$ acts without inversions on a systolic complex and fixes two vertices, then it fixes pointwise every combinatorial geodesic between them [10, Lemma 14]. Systolic complexes were first introduced by Chepoi under the name of bridged complexes in [7].…”

“…Since conjugacy stability is preserved by conjugation, the previous theorem solves the conjugacy stability problem for (2,2)‐free two‐dimensional Artin groups. Note that Theorem 1.4 is a generalization of [10, Theorem C].…”