What are the interesting applications of hyperbolic geometry? I am aware that, historically, hyperbolic geometry was useful in showing that there can be consistent geometries that satisfy the first 4 axioms of Euclid's elements but not the fifth, the infamous parallel lines postulate, putting an end to centuries of unsuccesfull attempts to deduce the last axiom from the first ones.
It seems to be, apart from this fact, of genuine interest since it was part of the usual curriculum of all mathematicians at the begining of the century and also because there are so many books on the subject.
However, I have not found mention of applications of hyperbolic geometry to other branches of mathematics in the few books I have sampled. Do you know any or where I could find them? 

It is my understanding that the principle application of hyperbolic geometry is in physics. Specifically in special relativity. The Lorentz group of Lorentz transformations is a non-compact hyperbolic manifold. But it also shows up in general relativity and astrophysics as the space surrounding black holes is hyperbolic(negative curvature).
