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**Fuchsian groups.**
*(English)*
Zbl 0753.30001

Chicago Lectures in Mathematics. Chicago: The University of Chicago Press,. x, 175 p. (1992).

This book is a graduate level introduction to the theory of Fuchsian groups, especially the geometric theory. It covers in a direct and uncomplicated fashion the basic material in hyperbolic geometry and the applications to establishing the fundamental properties of Fuchsian groups up to Jørgensen’s inequality. The theory of geometrically finite Fuchsian groups is treated separately and thoroughly, including a number of topics that are not easy to find in other accounts. Finally the author treats the theory of arithmetic Fuchsian groups in detail and this is very hard to find in a connected fashion elsewhere. This book would make an ideal text for individual study or for a graduate course. It is particularly welcome that the author has provided a set of exercises for the reader at the end of each chapter.

Reviewer: S.J.Patterson (Göttingen)

### MSC:

30-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functions of a complex variable |

30F35 | Fuchsian groups and automorphic functions (aspects of compact Riemann surfaces and uniformization) |

20H10 | Fuchsian groups and their generalizations (group-theoretic aspects) |

20-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory |