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

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 ${\mathrm{I}}_{\mathrm{\Gamma }}$-type in terms of skew braces when $\mathrm{\Gamma }$ is an Artin-Tits group of spherical type and classify all groups of ${\mathrm{I}}_{\mathrm{\Gamma }}$-type where $\mathrm{\Gamma }$ 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.