Discontinuous Groups of Isometries in the Hyperbolic Plane

In 1920’s and 1930’s, J. Nielsen published three long papers in Acta Mathematica on discontinuous groups of isometries of the non-euclidean plane. His further research in the field led him to an idea to describe the theory of discontinuous groups of isometries systematically and in full generality. After the Second World War, he started such a project together with W. Fenchel and they prepared the first version of the manuscript. W. Fenchel was able to continue (with his collaborators) the work on the project after J. Nielsen’s death and the typewritten version of the manuscript was ready at the end of the 80’s. The book under review is the final version of the manuscript, which was prepared for publication by A. L. Schmidt after W. Fenchel’s death in 1988. The book offers a systematic geometric treatment of discontinuous groups of isometries. It starts with a careful description of Möbius transformations of the Riemann sphere and their use in non-euclidean geometry. The second chapter contains a detailed description of discontinuous groups of motions of the unit discs with its hyperbolic metric. In the third chapter, associated surfaces and invariants needed for a classification are discussed. Elementary groups and elementary surfaces, the decompositions of the discontinuous groups and their normal forms, are all treated in the fourth chapter. The final chapter is devoted to isomorphisms of discontinuous groups, homeomorphisms of the corresponding discs and their extensions to the boundary. A specific feature of the book is that it is based entirely on geometric arguments. The Fenchel-Nielsen manuscript has been famous for a long time already and its final publication is a valuable edition to mathematical literature.

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