On the Borel complexity of the space of left-orderings of nilpotent groups

We present the first examples of nonabelian left-orderable groups for which the conjugacy orbit equivalence relation on their space of orders is smooth in the Borel sense, yet it has orbits of infinite length. All these examples are nilpotent groups. Moreover, we provide a sufficient condition to ensure that the space of orders of a finitely generated nilpotent group has a smooth conjugacy orbit relation. Finally, we provide a countable family of examples showing that nilpotence alone is not sufficient for smoothness.