Appears in collection : XVII International Luminy Workshop in Set Theory / XVII Atelier International de Théorie des Ensembles

A group is left-orderable if it admits a strict total order that is left-invariant under the group operation. The space of left-orderings of a given countable group is a well studied compact Polish space whose topological and dynamical features interact with the algebraic properties of the group. In this talk I will discuss the Borel complexity of the conjugacy equivalence relation on the spaces of left-orderings. This is joint work with Adam Clay.