Spaces of orders of some one-relator groups

TL;DR

This paper investigates the structure of orderable groups, demonstrating that certain one-relator groups and related constructions do not admit isolated left orders, using new perturbation techniques for group actions on the line.

Contribution

It introduces novel perturbation methods for analyzing group actions on the line and applies them to show the non-existence of isolated left orders in specific classes of groups.

Findings

Certain one-relator groups admit no isolated left orders

Perturbation techniques for group actions are developed and applied

Results extend understanding of orderability in complex group constructions

Abstract

We show that certain orderable groups admit no isolated left orders. The groups we consider are cyclic amalgamations of a free group with a general orderable group, the HNN extensions of free groups over cyclic subgroups, and a particular class of one-relator groups. In order to prove the results about orders, we develop perturbation techniques for actions of these groups on the line.