Book review

Everyone knows that Homer wrote the Iliad; many know the contents of this epic to a certain extent; but not so many have actually read it. There are moments in a mathematician's life when, partly because of the diversity of the areas of current mathematical research, partly because of the limited capacity of his brain, he finds himself in a similar position, and realizes with surprise how much other minds produced decades before he was born, and is amazed by the splendor of (some of) their achievements. Most readers of this review will have heard of wavelets, (super)strings, chaos and fractals, computer science or modular curves; but, I guess, there are quite few who would, for instance, be familiar with the theory of continuous fractions, or know how to derive some of the formulas in Bateman and Erdelyi's Higher Transcendental Functions. True, one cannot learn all the legacy of the "classics"; but there are some gems which deserve to be reminded to today's mathematical public, and it's a good idea to have books which would take care of this job. The book under review may qualify as one of them. Traditional hyperbolic geometry flourished in the second half of the last century, and although it contains results of undisputable depth and beauty, it is seldom included into university courses nowadays. On the other hand, it has stimulated the development of many areas which receive current interest and where an active research is going on.

The topics dealt with include: introduction to quadratic forms; the Klein and Poincaré models of hyperbolic geometry; trigonometry in the hyperbolic plane; Fuchsian groups (including Nielsen's theorem) and fundamental domains; uniformization, classification, and the space forms; Poincaré's theorem (and some combinatorial topology); hyperbolic 3-space, quaternions and Clifford groups; and there is an appendix about axioms for the plane geometry. The exposition is balanced and well written, and the book is beautifully typeset. There is an index, a short but adequate bibliography, and even a list of symbols. The only prerequisites needed are linear algebra and rudimentary point set topology, so the book will make an equally good reading for anyone from beginning graduate students to mature scientists.

There have been many people who have worked in hyperbolic geometry, and there have been equally many books written about it; but, as was said above, this does not detract from the present book any value. It will make a very pleasant contribution to any library, institutional or private; and its price makes buying an excellent bargain.