Following up on the previous MO question "Are there any important mathematical concepts without discrete analogue?", I'd like to ask the opposite: what are examples of notions in math that were not originally discrete, but have good discrete analogues? While a few examples arose in the answers to that earlier MO question, this wasn't what that question was asking, so I'm sure there are many more examples not mentioned there or at least not really explained there. What reminded me of this older MO question was seeing an MO question "Why is the Laplacian ubiquitous?", since that is an instance of an important notion which has a discrete analgoue.
In an answer, it would be interesting to hear about the relationship between the continuous and discrete versions of the notion, if possible, and references could also be helpful. Thanks!
Best Answer
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, 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.