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This is an introductory textbook on isometry groups of the
hyperbolic plane. Interest in such groups dates back more than 120
years. Examples appear in number theory (modular groups and
triangle groups), the theory of elliptic functions, and the theory
of linear differential equations in the complex domain (giving rise
to the alternative name Fuchsian groups). The current book is based
on what became known as the famous Fenchel-Nielsen manuscript.
Jakob Nielsen (1890-1959) started this project well before World
War II, and his interest arose through his deep investigations on
the topology of Riemann surfaces and from the fact that the
fundamental group of a surface of genus greater than one is
represented by such a discontinuous group. Werner Fenchel
(1905-1988) joined the project later and overtook much of the
preparation of the manuscript. The present book is special because
of its very complete treatment of groups containing reversions and
because it avoids the use of matrices to represent Moebius maps.
This text is intended for students and researchers in the many
areas of mathematics that involve the use of discontinuous groups.
The series is devoted to the publication of monographs and
high-level textbooks in mathematics, mathematical methods and their
applications. Apart from covering important areas of current
interest, a major aim is to make topics of an interdisciplinary
nature accessible to the non-specialist. The works in this series
are addressed to advanced students and researchers in mathematics
and theoretical physics. In addition, it can serve as a guide for
lectures and seminars on a graduate level. The series de Gruyter
Studies in Mathematics was founded ca. 35 years ago by the late
Professor Heinz Bauer and Professor Peter Gabriel with the aim to
establish a series of monographs and textbooks of high standard,
written by scholars with an international reputation presenting
current fields of research in pure and applied mathematics. While
the editorial board of the Studies has changed with the years, the
aspirations of the Studies are unchanged. In times of rapid growth
of mathematical knowledge carefully written monographs and
textbooks written by experts are needed more than ever, not least
to pave the way for the next generation of mathematicians. In this
sense the editorial board and the publisher of the Studies are
devoted to continue the Studies as a service to the mathematical
community. Please submit any book proposals to Niels Jacob. Titles
in planning include Flavia Smarazzo and Alberto Tesei, Measure
Theory: Radon Measures, Young Measures, and Applications to
Parabolic Problems (2019) Elena Cordero and Luigi Rodino,
Time-Frequency Analysis of Operators (2019) Mark M. Meerschaert,
Alla Sikorskii, and Mohsen Zayernouri, Stochastic and Computational
Models for Fractional Calculus, second edition (2020) Mariusz
Lemanczyk, Ergodic Theory: Spectral Theory, Joinings, and Their
Applications (2020) Marco Abate, Holomorphic Dynamics on Hyperbolic
Complex Manifolds (2021) Miroslava Antic, Joeri Van der Veken, and
Luc Vrancken, Differential Geometry of Submanifolds: Submanifolds
of Almost Complex Spaces and Almost Product Spaces (2021) Kai Liu,
Ilpo Laine, and Lianzhong Yang, Complex Differential-Difference
Equations (2021) Rajendra Vasant Gurjar, Kayo Masuda, and Masayoshi
Miyanishi, Affine Space Fibrations (2022)
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