The isomorphism problem for finitely generated bi-orderable groups

We analyze the classification problem for finitely generated orderable groups from the viewpoint of descriptive set theory. We analyze the standard Borel space of finitely generated left-orderable groups, and the subspace of finitely generated bi-orderable groups using spaces of relative cones. We use this setup to show that the isomorphism relation on finitely generated bi-orderable groups is weakly universal.