Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
From Groups to Geometry and Back
 
Vaughn Climenhaga University of Houston, Houston, TX
Anatole Katok Pennsylvania State University, University Park, PA
From Groups to Geometry and Back
Softcover ISBN:  978-1-4704-3479-3
Product Code:  STML/81
List Price: $59.00
Individual Price: $47.20
eBook ISBN:  978-1-4704-3753-4
Product Code:  STML/81.E
List Price: $49.00
Individual Price: $39.20
Softcover ISBN:  978-1-4704-3479-3
eBook: ISBN:  978-1-4704-3753-4
Product Code:  STML/81.B
List Price: $108.00 $83.50
From Groups to Geometry and Back
Click above image for expanded view
From Groups to Geometry and Back
Vaughn Climenhaga University of Houston, Houston, TX
Anatole Katok Pennsylvania State University, University Park, PA
Softcover ISBN:  978-1-4704-3479-3
Product Code:  STML/81
List Price: $59.00
Individual Price: $47.20
eBook ISBN:  978-1-4704-3753-4
Product Code:  STML/81.E
List Price: $49.00
Individual Price: $39.20
Softcover ISBN:  978-1-4704-3479-3
eBook ISBN:  978-1-4704-3753-4
Product Code:  STML/81.B
List Price: $108.00 $83.50
  • Book Details
     
     
    Student Mathematical Library
    Volume: 812017; 420 pp
    MSC: Primary 20; 51; Secondary 22; 54; 57;

    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.

  • Table of Contents
     
     
    • Chapters
    • Elements of group theory
    • Symmetry in the Euclidean world: Groups of isometries of planar and spatial objects
    • Groups of matrices: Linear algebra and symmetry in various geometries
    • Fundamental group: A different kind of group associated to geometric objects
    • From groups to geometric objects and back
    • Groups at large scale
  • Reviews
     
     
    • 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
  • Requests
     
     
    Review Copy – for publishers of book reviews
    Desk Copy – for instructors who have adopted an AMS textbook for a course
    Examination Copy – for faculty considering an AMS textbook for a course
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 812017; 420 pp
MSC: Primary 20; 51; Secondary 22; 54; 57;

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.

  • Chapters
  • Elements of group theory
  • Symmetry in the Euclidean world: Groups of isometries of planar and spatial objects
  • Groups of matrices: Linear algebra and symmetry in various geometries
  • Fundamental group: A different kind of group associated to geometric objects
  • From groups to geometric objects and back
  • Groups at large scale
  • 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
Review Copy – for publishers of book reviews
Desk Copy – for instructors who have adopted an AMS textbook for a course
Examination Copy – for faculty considering an AMS textbook for a course
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
You may be interested in...
Please select which format for which you are requesting permissions.