Parabolic subgroups of large-type Artin groups

María Cumplido Cabello; Alexandre Martin; Nicolas Vaskou

doi:10.1017/S0305004122000342

Parabolic subgroups of large-type Artin groups

María Cumplido Cabello, Alexandre Martin, Nicolas Vaskou

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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.

Original language	English
Pages (from-to)	393-414
Number of pages	22
Journal	Mathematical Proceedings of the Cambridge Philosophical Society
Volume	174
Issue number	2
Early online date	12 Sept 2022
DOIs	https://doi.org/10.1017/S0305004122000342
Publication status	Published - Mar 2023

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title = "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.",

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Parabolic subgroups of large-type Artin groups

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