## Geometry of Riemann surfaces and Teichmüller spaces.(English)Zbl 0917.32016

North-Holland Mathematics Studies. 169. Amsterdam etc: North-Holland. 263 p. (1992).
The book aims to study parameter spaces of Fuchsian groups representing Riemann surfaces in the spirit of the well-known Fricke-Klein moduli. The methods used are—as the authors themselves say—as elementary as possible, often boiling down to explicit computations of the multipliers or traces of Möbius transformations. Of course, everything depends on the classical uniformization theorem for Riemann surfaces (which is stated but not proved in the text). The book also discusses Fenchel-Nielsen coordinates and their degeneration in the case of noded Riemann surfaces. Advanced results on the construction and structure of the Teichmüller spaces are stated.
The chapter headings are: 1. Geometry of Möbius transformations; 2. Quasiconformal mappings; 3. Geometry of Riemann surfaces; 4. Moduli problems and Teichmüller spaces; 5. Moduli spaces. There are two appendices: A. Hyperbolic metric and Möbius groups; B. Traces of matrices.

### MSC:

 32G15 Moduli of Riemann surfaces, Teichmüller theory (complex-analytic aspects in several variables) 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces 30F60 Teichmüller theory for Riemann surfaces