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

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 Δ-product, which we study in generality. We also study the uniqueness of roots up to conjugacy in hosohedral-type groups.