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? 

I am not a mathematician, but my "Barron's dictionary of Mathematics Terms'  Second Edition
  1995 copyright edited by Douglas Dowing says one example of  hyperbolic function use is
"catenary";  catenary: study of the curve formed by a flexible rope hanging between two points. 
