Orderable groups

TL;DR

This paper explores the structure and classification of orderings in groups, introducing new characterizations, and analyzing the space of orderings in specific groups like free groups and Thompson's group, revealing complex topological properties.

Contribution

It provides a new characterization of the Conrad property, classifies groups with finitely many Conradian orderings, and describes the topology of the space of orderings in key groups.

Findings

Finitely many Conradian orderings imply finitely many left-orderings.

The space of left-orderings of free groups has a dense orbit and no isolated points.

The space of bi-orderings of Thompson's group contains isolated points and Cantor sets.

Abstract

In Chapter 1 we give the basic background and notations. We also give a new characterization of the Conrad property for orderings. In Chapter 2, we use the new characterization of the Conradian property to give a classification of groups admitting (only) finitely many Conradian orderings \S 2.1. Using this classification we deduce a structure theorem for the space of Conradian orderings \S 2.2. In addition, we are able to give a structure theorem for the space of left-orderings on a group by studying the possibility of approximating a given ordering by its conjugates \S 2.3. In Chapter 3 we show that, for groups having finitely many Conradian orderings, having an isolated left-ordering is equivalent to having only finitely many left-orderings. In Chapter 4, we prove that the space of left-orderings of the free group on $n\geq2$ generators have a dense orbit under the natural action of the free group on it. This gives a new proof of the fact that the space of left-orderings of the free group in at least two generators have no isolated point. In Chapter 5, we describe the space of bi-orderings of the Thompson's group $\efe$. We show that this space contains eight isolated points together with four canonical copies of the Cantor set.