Book indication on Hyperbolic space

Question

could someone tell me a good introductory book on hyperbolic space? A book that will get me well on the road to understanding CMC surfaces in the hyperbolic. I've looked for books on the internet, but I'm not sure if they are good references.

Answer 1

The optimal answer depends on your starting point, but one choice is the short elementary notes by Cannon et al.:

J.W. Cannon, et al. "Hyperbolic Geometry", Flavors of Geometry 31 (1997), 59-115.

It is freely available from MSRI.

From the introduction:

These notes are intended as a relatively quick introduction to hyperbolic geometry. They review the wonderful history of non-Euclidean geometry. They give very different analytic models for and several combinatorial approximations to non-Euclidean geometry by means of which the reader can develop an intuition for the behavior of this geometry. They develop a number of the properties of this geometry that are particularly important in topology and group theory. They indicate some of the fundamental problems being approached by means of non-Euclidean geometry in topology and group theory.