Negative curvature of Riemannian manifolds, originally a differentiable theory, has been discretized in several phases. The first phase might have been Dehn's algorithm for the word problem in a surface group; I am guessing that at the time this might have seemed more an "application" of hyperbolic geometry than a discretization of it. But then comes the next big phase, the development of [small cancellation theory][1], in which Dehn's algorithm (and related tools) were applied to many abstractly defined groups. The culminating phase was the development (by Gromov among others) of the theory of [hyperbolic groups][2].


  [1]: http://en.wikipedia.org/wiki/Small_cancellation_theory
  [2]: http://en.wikipedia.org/wiki/Hyperbolic_group