**Student Mathematical Library**

Volume: 81;
2017;
420 pp;
Softcover

MSC: Primary 20; 51;
Secondary 22; 54; 57

**Print ISBN: 978-1-4704-3479-3
Product Code: STML/81**

List Price: $58.00

Individual Price: $46.40

**Electronic ISBN: 978-1-4704-3753-4
Product Code: STML/81.E**

List Price: $58.00

Individual Price: $46.40

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#### Supplemental Materials

# From Groups to Geometry and Back

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*Vaughn Climenhaga; Anatole Katok*

Groups arise naturally as symmetries of
geometric objects, and so groups can be used to understand geometry
and topology. Conversely, one can study abstract groups by using
geometric techniques and ultimately by treating groups themselves as
geometric objects. This book explores these connections between group
theory and geometry, introducing some of the main ideas of
transformation groups, algebraic topology, and geometric group
theory.

The first half of the book introduces basic notions of group theory
and studies symmetry groups in various geometries, including
Euclidean, projective, and hyperbolic. The classification of
Euclidean isometries leads to results on regular polyhedra and
polytopes; the study of symmetry groups using matrices leads to Lie
groups and Lie algebras.

The second half of the book explores ideas from algebraic topology
and geometric group theory. The fundamental group appears as yet
another group associated to a geometric object and turns out to be a
symmetry group using covering spaces and deck transformations. In the
other direction, Cayley graphs, planar models, and fundamental domains
appear as geometric objects associated to groups. The final chapter
discusses groups themselves as geometric objects, including a gentle
introduction to Gromov's theorem on polynomial growth and Grigorchuk's
example of intermediate growth.

The book is accessible to undergraduate students (and anyone else)
with a background in calculus, linear algebra, and basic real
analysis, including topological notions of convergence and
connectedness.

This book is a result of the MASS course in algebra at Penn State
University in the fall semester of 2009.

This book is published in cooperation with Mathematics Advanced Study Semesters

#### Readership

Undergraduate and graduate students interested in group theory and geometry.

#### Reviews & Endorsements

Despite the beauty of the subject and the many applications to other areas of mathematics and physics, the geometry of group actions is not a common part of an undergraduate mathematics curriculum. The book under review attempts to fill that gap...The text is well written in a conversational style with many nice figures. It is a pleasure to read, for the instructor.

-- Cristopher H. Cashen, Mathematical Reviews

The clarity of the exposition and the richness of the topics make this a valuable addition to undergraduate math libraries.

-- J. McCleary, CHOICE

#### Table of Contents

# Table of Contents

## From Groups to Geometry and Back

- Cover Cover11
- Title page iii4
- Contents v6
- Foreword: MASS at Penn State University xi12
- Preface xiii14
- Guide for instructors xvii18
- Chapter 1. Elements of group theory 122
- Chapter 2. Symmetry in the Euclidean world: Groups of isometries of planar and spatial objects 6990
- Lecture 7. Isometries of \RR² and \RR³ 6990
- Lecture 8. Classifying isometries of \RR² 82103
- Lecture 9. The isometry group as a semidirect product 91112
- Lecture 10. Discrete isometry groups in \RR² 100121
- Lecture 11. Isometries of \RR³ with fixed points 114135
- Lecture 12. Finite isometry groups in \RR³ 121142
- Lecture 13. The rest of the story in \RR³ 133154
- Lecture 14. A more algebraic approach 143164

- Chapter 3. Groups of matrices: Linear algebra and symmetry in various geometries 155176
- Lecture 15. Euclidean isometries and linear algebra 155176
- Lecture 16. Complex matrices and linear representations 165186
- Lecture 17. Other geometries 176197
- Lecture 18. Affine and projective transformations 187208
- Lecture 19. Transformations of the Riemann sphere 197218
- Lecture 20. A metric on the hyperbolic plane 204225
- Lecture 21. Solvable and nilpotent linear groups 212233
- Lecture 22. A little Lie theory 221242

- Chapter 4. Fundamental group: A different kind of group associated to geometric objects 233254
- Chapter 5. From groups to geometric objects and back 269290
- Lecture 26. The Cayley graph of a group 269290
- Lecture 27. Subgroups of free groups via covering spaces 280301
- Lecture 28. Polygonal complexes from finite presentations 290311
- Lecture 29. Isometric actions on \HH² 302323
- Lecture 30. Factor spaces defined by symmetry groups 311332
- Lecture 31. More about 𝑆𝐿(𝑛,\ZZ) 327348

- Chapter 6. Groups at large scale 337358
- Lecture 32. Introduction to large scale properties 337358
- Lecture 33. Polynomial and exponential growth 348369
- Lecture 34. Gromov’s Theorem 356377
- Lecture 35. Grigorchuk’s group of intermediate growth 366387
- Lecture 36. Coarse geometry and quasi-isometries 376397
- Lecture 37. Amenable and hyperbolic groups 385406

- Hints to selected exercises 395416
- Suggestions for projects and further reading 401422
- Bibliography 409430
- Index 413434
- Back Cover Back Cover1442