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From Groups to Geometry and Back

Vaughn Climenhaga and Anatole Katok
American Mathematical Society
Publication Date: 
Number of Pages: 
Student Mathematical Library 81
[Reviewed by
Scott A. Taylor
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Vast amounts of incredibly interesting mathematics concern the relationship between geometry and algebra. Sadly undergraduates see very little of this terrain. Good courses in linear algebra and calculus give glimpses, but this is beautiful (and applicable!) mathematics that deserves an extended trek.

From Groups to Geometry and Back is a reasonable choice of guide for such a journey. The hike, however, is a challenge: if you’re going to go with this guide, you’ll need to be in shape! Readers will need a solid background in algebra and some form of topology or analysis. All the needed definitions are included in the book, along with extremely helpful remarks to help guide intuition. The pace is, however, quite breath-taking. In the course of almost 400 pages we are taken from the definition of “group” through lots of different geometries such as 3-dimensional euclidean space, 2-dimensional hyperbolic geometry, projective geometry. We see many interesting examples: group actions on various geometric spaces, the quotient spaces resulting from these group actions, and various classification results concerning what algebraic properties of a group enable actions on certain kinds of spaces. Throughout significant use is made of linear algebra (matrix representations, eigenvalues, inner products). The excursion ends with an introduction to geometric group theory via discussion of the relationships between the growth rate of a group and its algebraic and geometric structure.

This book began life as lecture notes for a course at Penn State’s Mathematics Advanced Study Semester (MASS). I suspect that outside of a similarly intense environment the book would be far too advanced for almost all undergraduates, including those in most major capstone courses. The authors recognize this and give advice on how the book might be adapted situations other than MASS. The writing, organization, and vision is so tight, however, that I think it would probably be better to assign a text more closely aligned with whatever the course is.

In its defense, From Groups to Geometry and Back is beautifully written and organized and would make great reading for students in the summer between getting their undergraduate degree and mathematics graduate school. Additionally, for those of us who teach aspects of this material in algebra, topology, or geometry courses, we can definitely make use of the lovely explanations of many of the results we might cover. Furthermore, the book outlines a vision of the relationship between geometry and algebra which deserved broader exposure. The unifying perspective of the relationship between matrix groups and geometries is one that could even be communicated to students in a linear algebra course.

The book should have included more extensive references to the published literature and previous expositions. It draws on the work of a lot of mathematicians from the previous two centuries and it would be nice to see that reflected in the references.


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Scott Taylor is an associate professor at Colby College. He’s always looking for more ways to get geometry and topology into the undergraduate curriculum.

See the table of contents in the publisher's webpage.