Conjugacy invariants and rigidity in Garside groups: a uniformity phenomenon

2025/09/01 by Matthieu Calvez, Calvez, Matthieu, Garnier, Owen +4
Computer Science · #Contact Mechanics and Variational Inequalities #FOS: Mathematics #Group Theory (math.GR)

paper · pdf · doi:10.48550/arxiv.2509.01361

openalex publication_date 2025/09/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29

Abstract

Consider an element~x of a Garside group which is rigid in the sense of Garside-theory. Let SC(x) be the set of rigid conjugates of~x -- this is a well-known characteristic subset of the conjugacy class of~x. We present computational evidence that the sequence ( |SC(xⁿ)| )_{n∈\mathbb N} is not only bounded, but in fact periodic, and that there is a bound on the length of the period which depends only on the underlying group and its Garside structure. We prove this result in the special case of the circular Garside groups, including the 2-generator Artin groups with their classical and dual structures (where we prove that the sequence is always constant), and in the case of the dual 4-strand braid group.

Citations

• Constructing metric spaces from systems of walls
• Uniqueness of roots up to conjugacy in circular and hosohedral-type Garside groups
• Morse subsets of injective spaces are strongly contracting
• What is a hierarchically hyperbolic space?
• Garside and quadratic normalisation: a survey
• The development version of the CHEVIE package of GAP3
• A fast solution to the conjugacy problem in the 4-strand braid group
• The cyclic sliding operation in Garside groups
• Conjugacy in Garside groups II: Structure of the ultra summit set
• A New Approach to the Conjugacy Problem in Garside Groups
• On the structure of the centralizer of a braid
• Gaussian Groups and Garside Groups, Two Generalisations of Artin Groups
• A new approach to the word and conjugacy problems in the braid groups
• ALGORITHMS FOR POSITIVE BRAIDS
• Artin-Gruppen und Coxeter-Gruppen

Related

• Conjugacy in Garside groups I: Cyclings, powers, and rigidity
• Conjugacy in Garside Groups III: Periodic braids
• Notes on periodic elements of Garside groups
• Conjugacy in Garside groups II: Structure of the ultra summit set
• Conjugacy problem for braid groups and Garside groups