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

TL;DR

This paper investigates the Borel complexity of the space of left-orderings in nilpotent groups, providing examples and conditions for when the conjugacy orbit relation is smooth, revealing nuanced differences in nilpotent group orderings.

Contribution

It presents the first examples of nonabelian nilpotent groups with smooth conjugacy orbit relations despite infinite orbits, and offers a sufficient condition for smoothness in such spaces.

Findings

Examples of nilpotent groups with smooth orbit relations despite infinite orbits

A sufficient condition for the space of orders to have a smooth conjugacy orbit relation

Demonstration that nilpotence alone does not guarantee smoothness

Abstract

We give the first examples of nonabelian left-orderable groups such that the conjugacy orbit equivalence relation on its space of orders has infinity orbits, yet it is smooth in the Borel sense. The examples are all nilpotent groups and we provide a sufficient condition so that the space of orders of a nilpotent group has a smooth conjugacy orbit relation. We also show with different examples that nilpotence is not a sufficient condition for smoothness.