Ordered groups and topology
Adam Clay, Dale Rolfsen

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…
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Taxonomy
TopicsGeometric and Algebraic Topology · Mathematical Dynamics and Fractals · Homotopy and Cohomology in Algebraic Topology
