Condensation and left-orderable groups

In this talk we shall discuss condensed points in the Polish space of left-orderings of a fixed left-orderable groups. We will describe new techniques to show that the conjugacy relation on the space of left-orderings is not smooth. We discuss how these methods apply to a large class of left-orderable groups, and they shed light on spaces of left-orderings with low Borel complexity. This is joint work with Adam Clay.