Ordered groups and topology

TL;DR

This book explores the deep connections between topology and orderable groups, covering algebraic properties, topological methods, and applications to 3-manifolds, braid groups, and homeomorphism groups, highlighting recent advances and open problems.

Contribution

It provides new proofs and insights into the structure of orderings on groups, including a novel proof of the equivalence of local indicability and Conradian orderings, and analyzes the space of all orderings.

Findings

Orderings of important groups in topology are explicitly constructed.

The space of left-orderings of a group is either finite or uncountably infinite.

New proofs of classical theorems regarding orderability and the structure of ordering spaces.

Abstract

This is a draft of a book submitted for publication by the AMS. Its theme is the remarkable interplay, accelerating in the last few decades, between topology and the theory of orderable groups, with applications in both directions. It begins with an introduction to orderable groups and their algebraic properties. Many of the algebraic results are proved by topological methods, via consideration of the space of orderings. After a discussion H\"older's theorem and some dynamical aspects of orderable groups, we provide explicit orderings of important groups in topology: free groups and most surface groups. Next we consider orderability of the fundamental groups of three-dimensional manifolds. All knot groups, and more generally groups of 3-manifolds with positive first Betti number are left-orderable, in fact locally indicable and sometimes even bi-orderable. However when the first homology is finite the situation is more subtle, and we find connections with foliations, branched coverings, surgery, and conjecturally Heegaard-Floer homology. Braid groups are considered in some detail, including Dehornoy's ordering of the braid groups and its later generalizations due to Thurston. This is followed by a discussion of recent applications of Dehornoy's ordering to knot theory. A short chapter outlines a proof that the group of PL homeomorphisms of the $n$-dimensional cube (fixed on the boundary) is left-orderable; a property conjectured to be true for the group of homeomorphisms in dimension two. We present a new proof that local indicability of a group is equivalent to the existence of a "Conradian" left-ordering. A final chapter considers the space $LO(G)$ of left-orderings of a group $G$. We give a new proof of Sikora's theorem that $LO(\mathbb{Z}^n), n >1$ is a Cantor set, as well as a proof of Linnell's theorem that for any group $G$, $LO(G)$ is either finite or uncountable.