Free products of circularly-ordered groups with amalgamated subgroup

TL;DR

This paper characterizes when free products with amalgamation of circularly-ordered groups can be circularly ordered, using a categorical framework and existing results on left-ordered central extensions.

Contribution

It provides necessary and sufficient conditions for such amalgamated free products to admit a compatible circular ordering, extending prior work through a new categorical approach.

Findings

Established a categorical framework for the problem

Derived necessary and sufficient conditions for circular orderings

Connected the problem to amalgamating left-ordered central extensions

Abstract

This paper gives necessary and sufficient conditions that the free product with amalgamation of circularly-ordered groups admit a circular ordering extending the given orderings of the factors. Our result follows from establishing a categorical framework that allows the problem to be restated in terms of amalgamating certain left-ordered central extensions, where we are able to apply work of Bludov and Glass.