Topics in the theory of Riemann surfaces / Robert D. M. Accola

This book addresses those readers with a familiarity with the known texts about compact Riemann surfaces (e.g. the books of Ahlfors-Sario, Gunning, Forster), who are also looking to quickly learn about the nearest results in the area.

After a short but very useful review of the basic results in the theory of compact Riemann surfaces, the author considers Weierstras points and their generalizations. These concepts are then applied to present a series of older as well as more recent results about groups of automorphisms of compact Riemann surfaces. This is done by a close examination of “exceptional” fixed points of automorphisms, which justifies the title of the sixth chapter: “When are fixed points of automorphisms exceptional in some other sense?”

The large number of concrete questions and examples makes this book very useful as well as readable. The beginner is lead directly to questions that stimulate his imagination. The text is also very useful for students who are looking for ideas for an excellent thesis. For an eventual second edition, the author might want to consider enlarging the bibliography; some important papers of Japanese mathematicians are not included. To reiterate, in only 100 pages, the author provides a nice presentation of the modern development of the theory of compact Riemann surfaces as well as suggestions for further investigation. This book will be useful for both the beginner as well as the more experienced researcher. (Zentralblatt)