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Low Dimensional Geometry: From Euclidean Surfaces to Hyperbolic Knots

Francis Bonahon
American Mathematical Society/intitute for Advanced Study
Publication Date: 
Number of Pages: 
Student Mathematical Library 49
[Reviewed by
William J. Satzer
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The Student Mathematical Library series, published by the American Mathematical Society, includes a number of attractive texts aimed at undergraduates and designed to offer challenges above and beyond the usual curriculum. In many cases the books attempt to take the student from a very basic level to the edge of current research. The current book aims to introduce students to some of Thurston’s striking developments in three-dimensional geometry – in particular, the surprising appearance of hyperbolic geometry in a purely topological problem.

The first two-thirds of the book is about two-dimensional geometry, focusing mostly on the hyperbolic plane and introducing the basic elements of hyperbolic geometry. Very quickly the author moves to the construction of locally homogeneous spaces by gluing the sides of a polygon. This in turn leads to an investigation of the tessellations associated with such constructions. One extraordinary example is the gorgeous Farey tessellation of the hyperbolic plane. By bending a Farey tessellation we are led to three-dimensional hyperbolic geometry and a discussion of Kleinian and Fuchsian groups.

Using one particular example of a Farey tessellation whose tiling group is a Kleinian group we see an unexpected connection with a figure-eight knot and its complement. The author then moves on to generalize the hyperbolic metric on the figure-eight knot complement and brings us first to the geometrization theorem for knot complements. Then he offers a quick look at Thurston’s general geometrization theorem.

Although the book begins at a relatively elementary level, it moves pretty quickly into much more challenging territory. Although, in principle, not much more than a bit of multivariable calculus is needed, both the material and the level of rigor require an additional level of sophistication. The author makes an odd choice early on when he decides that the customary treatment of cut-and-paste descriptions of the quotient topology is not adequate. He decides instead to develop quotient metric spaces and a quotient semi-metric (in full generality) because he finds that more intuitive. The cost is that proofs become more technical. The author regards this as a good thing, an opportunity to develop technical skills in rigorous proofs. This seems to me an unnecessary complication given the other challenges that students face here. It’s also arguable that quotient metric spaces make for a more intuitive approach.

Throughout the book the author provides useful indications in the text for sections that can be skipped on first reading. This is especially important here because of the mass of technical details that sometimes threaten to overwhelm the developing story. There are also many good exercises, coordinated well with the text, and a very useful bibliography.

Bill Satzer ( is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.