About left orders in Garside groups

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.

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References in corpus (3)

• Amenable groups that act on the line
• Finite quotients of groups of I-type
• Coxeter-like groups for groups of set-theoretic solutions of the Yang--Baxter equation

Cited by in corpus (9)

• On structure groups of set-theoretic solutions to the Yang-Baxter equation
• Factorizations of skew braces
• A characterization of finite multipermutation solutions of the Yang-Baxter equation
• Set-theoretic solutions to the Yang-Baxter equation and generalized semi-braces
• Decomposition theorems for involutive solutions to the Yang-Baxter equation
• Combinatorial solutions to the reflection equation
• Retractability of solutions to the Yang-Baxter equation and $p$-nilpotency of skew braces
• The Yang-Baxter equation and Thompson's group $F$
• Skew braces: a brief survey