Groups of $\mathrm{I}_G$-type

TL;DR

This paper broadens the concept of groups of I_G-type for Garside groups, characterizes them using skew braces, and classifies all such groups for irreducible spherical Artin-Tits groups, including braid groups.

Contribution

It introduces a generalized notion of I_G-type groups for left-ordered groups and provides a classification for irreducible spherical Artin-Tits groups.

Findings

Characterization of I_G-type groups via skew braces.

Classification of I_G-type groups for irreducible spherical Artin-Tits groups.

Answer to Dehornoy et al.'s question on B_n structures.

Abstract

In this work, we address a question posed by Dehornoy et al. in the book "Foundations of Garside Theory" that asks for a theory of groups of $\mathrm{I}_G$-type when $G$ is a Garside group. In this article, we introduce a broader notion than the one suggested by Dehornoy et al.: given a left-ordered group $G$, we define a group of $\mathrm{I}_G$-type as a left-ordered group whose partial order is isomorphic to those of $G$. Furthermore, we develop methods to give a characterization of groups of $\mathrm{I}_{\Gamma}$-type in terms of skew braces when $\Gamma$ is an Artin-Tits group of spherical type and classify all groups of $\mathrm{I}_{\Gamma}$-type where $\Gamma$ is an irreducible spherical Artin-Tits group, therefore providing an answer to another question of Dehornoy et al. concerning $\mathrm{I}_{B_n}$ structures where $B_n$ is the braid group on $n$ strands with its canonical Garside structure.