This book presents topics in 2-dimensional hyperbolic geometry. The approach is to define metrics from an infinitesimal point of view; first the density is defi... » 

This book presents topics in 2-dimensional hyperbolic geometry. The approach is to define metrics from an infinitesimal point of view; first the density is defined and then the metric via integration. The study of hyperbolic geometry in arbitrary domains requires the concepts of surfaces and covering spaces as well as uniformization and Fuchsian groups. New material on hyperbolic and hyperbolic-like metrics is presented. These are generalizations of the Kobayashi and Caratheodory metrics for plane domains. The book concludes with applications to holomorphic dynamics including new results and accessible open problems. «