About left orders in Garside groups

TL;DR

This paper investigates the orderability properties of structure groups arising from solutions to the quantum Yang-Baxter equation, revealing diverse order structures including non-bi-orderability and complex spaces of left orders.

Contribution

It demonstrates that these Garside groups are not bi-orderable and explores the rich variety of their left orderings, including Cantor set homeomorphisms and non-unique product structures.

Findings

The groups are not bi-orderable.

Existence of a Cantor set of left orders.

Presence of non-unique product structures.

Abstract

We consider the structure group of a non-degenerate symmetric (non-trivial) set-theoretical solution of the quantum Yang-Baxter equation. This is a Bieberbach group and also a Garside group. We show this group is not bi-orderable, that is it does not admit a total order which is invariant under left and right multiplication. Regarding the existence of a left invariant total ordering, there is a great diversity. There exist structure groups with space of left orders homeomorphic to the Cantor set and all left orders Conradian, while there exist others that are even not unique product groups.