Borel structures on the space of left-orderings

Abstract: In this paper we study the Borel structure of the space of left-orderings $\mathrm{LO}(G)$ 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 $\mathrm{LO}(G)/G$ 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 $\mathbb{F}{2}$ on $\mathrm{LO}(\mathbb{F}{2})$ is universal, and leverage this result to provide many other examples of countable left-orderable groups $G$ such that the natural $G$-action on $\mathrm{LO}(G)$ induces a universal countable Borel equivalence relation.