Orderable groups

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.