A note on geometric constructions of bi-invariant orderings

TL;DR

This paper introduces a geometric method to construct bi-invariant orderings of residually torsion-free nilpotent groups using Chen's iterated integrals, linking orderings with rational homotopy theory.

Contribution

It generalizes the Magnus ordering of free groups through a geometric approach involving Chen's integrals and classical central extension techniques.

Findings

Constructs bi-invariant orderings via Chen's iterated integrals.

Establishes a connection between bi-orderings and rational homotopy theory.

Provides a unified geometric framework for known orderings.

Abstract

We construct bi-invariant total orderings of residually torsion-free nilpotent groups by using Chen's iterated integrals. This construction can be seen as a generalization of the Magnus ordering of the free groups, and equivalent to the classical construction which uses an iteration of central extensions. Our geometric construction provides a connection between bi-orderings and the rational homotopy theory.