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 define the space of finitely generated left-orderable groups, and the subspace of finitely generated bi-orderable groups using spaces of relative cones, and show that both spaces are Polish. We use this setup to show that the isomorphism relation on the space of finitely generated bi-orderable groups is weakly universal.