Borel structures on the space of left-orderings

In this paper we study the Borel structure of the space of left-orderings $\mathrm{L}\mathrm{O}(G)$ of a group $G$ 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 $\mathrm{L}\mathrm{O}(G)\mathrm{/}G$ 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 ${\mathbb{F}}_2$ on $\mathrm{L}\mathrm{O}({\mathbb{F}}_2)$ is universal, and leverage this result to provide many other examples of countable left-orderable groups $G$ such that the natural $G$-action on $\mathrm{L}\mathrm{O}(G)$ induces a universal countable Borel equivalence relation.