The space of bi-invariant orders on a nilpotent group

TL;DR

This paper investigates the structure and properties of the space of bi-invariant orders on torsion-free, nonabelian, nilpotent groups, revealing its topological features and group actions, including non-faithful automorphism actions.

Contribution

It establishes that the space of bi-invariant orders has no isolated points, explores the faithfulness of automorphism and commensurator group actions, and connects to recent work on automorphism group actions.

Findings

The space of bi-invariant orders is a Cantor set if the group is countable.

The outer automorphism group does not always act faithfully on this space.

The abstract commensurator group acts faithfully on the space of left-invariant orders.

Abstract

We prove a few basic facts about the space of bi-invariant (or left-invariant) total order relations on a torsion-free, nonabelian, nilpotent group G. For instance, we show that the space of bi-invariant orders has no isolated points (so it is a Cantor set if G is countable), and give examples to show that the outer automorphism group of G does not always act faithfully on this space. Also, it is not difficult to see that the abstract commensurator group of G has a natural action on the space of left-invariant orders, and we show that this action is faithful. These results are related to recent work of T.Koberda that shows the automorphism group of G acts faithfully on this space.