Uniqueness of roots up to conjugacy in circular and hosohedral-type Garside groups

TL;DR

This paper proves the uniqueness of roots up to conjugacy in circular and hosohedral-type Garside groups, leading to classification results and insights into complex braid groups of rank 2.

Contribution

It introduces and analyzes hosohedral-type Garside groups, extending root conjugacy results and classification to these new group classes.

Findings

Roots are unique up to conjugacy in circular groups.

Complete classification of circular groups up to isomorphism.

Roots are unique up to conjugacy in complex braid groups of rank 2.

Abstract

We consider a particular class of Garside groups, which we call circular groups. We mainly prove that roots are unique up to conjugacy in circular groups. This allows us to completely classify these groups up to isomorphism. As a consequence, we obtain the uniqueness of roots up to conjugacy in complex braid groups of rank 2. We also consider a generalization of circular groups, called hosohedral-type groups. These groups are defined using circular groups, and a procedure called the Delta-product, which we study in generality. We also study the uniqueness of roots up to conjugacy in hosohedral-type groups.