Appears in collection : 2024 - T2 - WS1 - Low Dimensional Actions

In this talk we will discuss the problem of determining the Borel complexity of the space of left-orders LO(G) of a countable left-orderable group G modulo the conjugacy G-action. We will see how this problem is connected to some well-studied topological properties of LO(G) such as the existence of dense orbits, and condensed orders. We will give an overview of our results showing that certain groups have nonstandard orbit space LO(G)/G. Time permitting, we will list open problems and discuss future directions.

Most of the results presented are joint work with Adam Clay.