**Graduate Studies in Mathematics**

Volume: 176;
2016;
154 pp;
Hardcover

MSC: Primary 20; 57;

Print ISBN: 978-1-4704-3106-8

Product Code: GSM/176

List Price: $79.00

AMS Member Price: $63.20

MAA member Price: $71.10

**Electronic ISBN: 978-1-4704-3562-2
Product Code: GSM/176.E**

List Price: $79.00

AMS Member Price: $63.20

MAA member Price: $71.10

#### You may also like

#### Supplemental Materials

# Ordered Groups and Topology

Share this page
*Adam Clay; Dale Rolfsen*

This book deals with the connections between
topology and ordered groups. It begins with a self-contained
introduction to orderable groups and from there explores the
interactions between orderability and objects in low-dimensional
topology, such as knot theory, braid groups, and 3-manifolds, as well
as groups of homeomorphisms and other topological structures. The
book also addresses recent applications of orderability in the studies
of codimension-one foliations and Heegaard-Floer homology. The use of
topological methods in proving algebraic results is another feature of
the book.

The book was written to serve both as a textbook for graduate
students, containing many exercises, and as a reference for
researchers in topology, algebra, and dynamical systems. A basic
background in group theory and topology is the only prerequisite for
the reader.

#### Readership

Graduate students and researchers interested in low-dimensional topology, 3-manifolds, and knot theory.

#### Reviews & Endorsements

The book finds a good balance between being a resource for researchers and a graduate textbook.

-- Sebastian Wolfgang Hensel, Mathematical Reviews

The diligent and disciplined reader of this thin (<150 pp) book will be rewarded by a lot more than knowledge of ordered groups.

-- Lee P. Neuwirth, Zentralblatt MATH

Given the huge popularity enjoyed by low dimensional topology these days, and all for good reason, it should make a very positive impact. The book is easy to read and deals with very pretty mathematics.

-- Michael Berg, MAA Reviews

#### Table of Contents

# Table of Contents

## Ordered Groups and Topology

- Cover Cover11
- Title page iii4
- Contents v6
- Preface ix10
- Chapter 1. Orderable groups and their algebraic properties 112
- 1.1. Invariant orderings 213
- 1.2. Examples 314
- 1.3. Bi-orderable groups 617
- 1.4. Positive cone 718
- 1.5. Topology and the spaces of orderings 819
- 1.6. Testing for orderability 1122
- 1.7. Characterization of left-orderable groups 1324
- 1.8. Group rings and zero divisors 1526
- 1.9. Torsion-free groups which are not left-orderable 1627

- Chapter 2. Hölder’s theorem, convex subgroups and dynamics 2132
- Chapter 3. Free groups, surface groups and covering spaces 3142
- Chapter 4. Knots 4354
- Chapter 5. Three-dimensional manifolds 6576
- Chapter 6. Foliations 7788
- Chapter 7. Left-orderings of the braid groups 91102
- Chapter 8. Groups of homeomorphisms 115126
- Chapter 9. Conradian left-orderings and local indicability 121132
- Chapter 10. Spaces of orderings 131142
- 10.1. The natural actions on 𝐿𝑂(𝐺) 132143
- 10.2. Orderings of \Zⁿ and Sikora’s theorem 133144
- 10.3. Examples of groups without isolated orderings 135146
- 10.4. The space of orderings of a free product 137148
- 10.5. Examples of groups with isolated orderings 139150
- 10.6. The number of orderings of a group 140151
- 10.7. Recurrent orderings and a theorem of Witte-Morris 144155

- Bibliography 147158
- Index 153164
- Back Cover Back Cover1167