This book contains a careful presentation of hyperbolic geometry based on an infinitesimal approach to the notion of a distance between two points. The principal notion here is the notion of a density. It makes it possible to measure lengths of curves, hence also the distances between points. After the motivation coming from Euclidean geometry, the hyperbolic metric on the unit disc in the complex plane is studied in detail. In the next two chapters, necessary notions from topology (covering spaces and the uniformization theorem) and from complex function theory (the Riemann mapping theorem and the Schwarz reflection principle) are introduced. After a discussion of the symmetry groups of the Euclidean and hyperbolic planes, densities for hyperbolic geometry of a domain, their generalizations (the Kobayashi metric and the Carathéodory pseudo-metric) and relations between holomorphic mappings and the hyperbolic metric are discussed. The next three chapters contain applications (iterated function systems and their limiting behaviour). The last chapters contain various estimates on hyperbolic metrics using inclusion mappings. Parts of the book can be used for lecture series of various lengths.

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