This book appears in the Springer Undergraduate Mathematics Series (SUMS) and provides an introduction to the geometry of hyperbolic plane, which has been for the past two centuries — and still is — an active field of mathematical research.
The book is self-contained and intended as a textbook or for a self-study/reading course. The author states that he wrote for a "third or fourth year" undergraduate student. This is very true, provided that the student has taken a sequence of Calculus courses, some Real Analysis, some Abstract Algebra, and has some knowledge of Complex Analysis (maybe a full course in Complex Analysis is not necessary, but it would be helpful…)
A course taught by the author at the University of Southampton in late 90s, as well as suggestions from other mathematicians and from students (both from the University of Southampton and from Rice University), have helped to produce a very good book: very readable, very well organized, with full proofs, with many examples, and with many exercises (the solutions to all the exercises are also included.) The author indicates that he has not made many changes in this second edition of the book, except in one chapter… and he has completely rewritten another chapter.
In short, I would recommend this book for consideration for anybody teaching a course in hyperbolic geometry. I would venture to say that even the beginning graduate students interested in the subject could benefit from reading this book.
Mihaela Poplicher is an assistant professor of mathematics at the University of Cincinnati. Her research interests include functional analysis, harmonic analysis, and complex analysis. She is also interested in the teaching of mathematics. Her email address is Mihaela.Poplicher@uc.edu.
Preamble to the Second Edition
Preamble to the First Edition
The Basic Spaces
The General Möbius Group
Length and Distance in H
Planar Models of the Hyperbolic Plane
Convexity, Area and Trigonometry
Solutions to Exercises
List of Notation