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- Publisher
- Wiley
- Copyright
- © 2022 London Mathematical Society
- ISSN
- 0024-6093
- eISSN
- 1469-2120
- DOI
- 10.1112/blms.12559
- Publisher site
- See Article on Publisher Site
Abstract
In this paper, we study the Borel structure of the space of left‐orderings LO(G)$\operatorname{LO}(G)$ of a group G$G$ modulo the natural conjugacy action, and by using tools from descriptive set theory, we find many examples of countable left‐orderable groups such that the quotient space LO(G)/G$\operatorname{LO}(G)/G$ is not standard. This answers a question of Deroin, Navas, and Rivas. We also prove that the countable Borel equivalence relation induced from the conjugacy action of F2$\mathbb {F}_{2}$ on LO(F2)$\operatorname{LO}(\mathbb {F}_{2})$ is universal, and leverage this result to provide many other examples of countable left‐orderable groups G$G$ such that the natural G$G$‐action on LO(G)$\operatorname{LO}(G)$ induces a universal countable Borel equivalence relation.
Journal
Bulletin of the London Mathematical Society – Wiley
Published: Feb 1, 2022