Orders on trees and free products of left-ordered groups

TL;DR

This paper introduces a method to construct total orders on the vertices of oriented trees using local up-down counts and geodesic edges, and applies it to prove that free products of left-orderable groups are left-orderable.

Contribution

It provides a new construction of vertex orders on trees and offers a concise proof of the left-orderability of free products of left-orderable groups.

Findings

Constructed total orders on tree vertices based on local counts.

Proved free products of left-orderable groups are left-orderable.

Simplified proof of Vinogradov's result using Bass-Serre theory.

Abstract

We construct total orders on the vertex set of an oriented tree. The orders are based only on up-down counts at the interior vertices and the edges along the unique geodesic from a given vertex to another. As an application, we provide a short proof (modulo Bass-Serre theory) of Vinogradov's result that the free product of left-orderable groups is left-orderable.