Left orders in Garside groups

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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 multiplications. Regarding the existence of a left invariant total ordering, there is a great diversity. There exist structure groups with a recurrent left order and with space of left orders homeomorphic to the Cantor set, while there exist others that are even not unique product groups.

Original languageEnglish
Pages (from-to)1349-1359
Number of pages11
JournalInternational Journal of Algebra and Computation
Issue number7
StatePublished - 1 Nov 2016

Bibliographical note

Publisher Copyright:
© 2016 World Scientific Publishing Company.


  • Garside groups
  • local indicability
  • orderability of groups
  • set-theoretical solutions of the quantum Yang-Baxter equation
  • unique product property

ASJC Scopus subject areas

  • General Mathematics


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