**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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# 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

#### Table of Contents

# Table of Contents

## From Groups to Geometry and Back

#### Readership

Undergraduate and graduate students interested in group theory and geometry.