Fundamental techniques in the study of parabolic subgroups of Artin groups

2025/09/10 by María Cumplido, Cumplido, María
Mathematics · #Algebraic Geometry and Number Theory #FOS: Mathematics #Finite Group Theory Research #Geometric and Algebraic Topology #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2509.08382

openalex publication_date 2025/09/10 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This survey was written on the occasion of the course I gave at the Winterbraids XIV workshop in Bordeaux (2025). Its main purpose is to present the techniques that have proven most effective in the study of parabolic subgroups of Artin groups, with particular emphasis on the parabolic subgroups intersection problem. The survey highlights the core ideas and strategies behind them, aiming to give the reader a concise and accessible entry point to the essential methods.

Citations

• On Artin groups admitting retractions to parabolic subgroups
• Intersection of Parabolic Subgroups in Euclidean Braid Groups: a short proof
• Parabolic subgroups of complex braid groups
• Intersection of parabolic subgroups in even Artin groups of FC-type
• Parabolic subgroups inside parabolic subgroups of Artin groups
• On parabolic subgroups of Artin groups
• Parabolic subgroups of two-dimensional Artin groups and systolic-by-function complexes
• Parabolic subgroups of large-type Artin groups
• Artin groups of infinite type: trivial centers and acylindical hyperbolicity
• Artin groups of euclidean type
• Parabolic and Quasiparabolic Subgroups of Free Partially Commutative Groups
• Artin groups of spherical type up to isomorphism
• The infimum, supremum and geodesic length of a braid conjugacy class
• Gaussian Groups and Garside Groups, Two Generalisations of Artin Groups
• ALGORITHMS FOR POSITIVE BRAIDS
• Artin-Gruppen und Coxeter-Gruppen

Related

• On parabolic subgroups of Artin groups
• Intersection of parabolic subgroups in even Artin groups of FC-type
• On Artin groups admitting retractions to parabolic subgroups
• On parabolic subgroups of Artin-Tits groups
• Parabolic subgroups of Artin groups of FC type