Hyperbolic geometry from a local viewpoint , by Linda Keen and Nikola

In addition to being packed with fundamental material important for every beginner in complex analysis, in expeditious and intuitive terms this little book transports the reader through a range of interesting topics in one-dimensional hyperbolic geometry, discrete subgroups, holomorphic dynamics and iterated function systems. In chapter after chapter, one quickly arrives at open problems and areas for research. The first half of the book carries the reader through the essential elements of Riemann surface theory, starting with geometry in the Euclidean plane and the Riemann sphere and going on to hyperbolic geometry in the hyperbolic plane. Topics include basic properties of holomorphic and univalent functions, Schwarz’s lemma, covering spaces, universal covering spaces, fundamental groups, discontinuous groups, Fuchsian groups. There is a very nice exposition of the Poincaré polygon theorem, which provides a sufficient condition for a subgroup of PSL(2,R) generated by side-pairings of a hyperbolic polygon to form a discrete group. There is also a concise and elementary exposition of the collar lemma. The collar lemma is a fundamental lemma for the analysis of hyperbolic surfaces. It provides a collar of definite thickness containing any closed geodesic on a hyperbolic surface, and as the geodesic shortens the collar gets thicker. One of its consequences is that there is a fixed lower bound on the lengths of intersecting closed geodesics. The lower bound is universal for all surfaces and all geodesics, and in this respect the lemma resembles the Heisenberg uncertainty principle. In Bers [4, page 443-449] one can find a history of the lemma and references for it. In the second half of the book, starting with the chapters on Kobayashi and Carathéodory metrics for hyperbolic plane domains, there are great numbers of new theorems, many of which are only recently proved and some of which appear for the first time in this publication. In this part the approach is more categorical. The basic objects of study are canonical procedures for defining conformal metrics on Riemann surfaces. The chief constraint is that such procedures are required to give metrics that satisfy the conclusion of Schwarz’s lemma. If X is a Riemann surface and we denote by σX the infinitesimal form of the canonically associated metric, we require that for any holomorphic map f mapping X into another Riemann surface Y we have the inequality