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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?

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    There are others on this site who are far more capable of giving a pertinent answer to this than me. While you're waiting for one to appear, you could have a glance at the Wikipedia page on the [modular group](http://en.wikipedia.org/wiki/Modular_group) and its [relationship to hyperbolic geometry](http://en.wikipedia.org/wiki/Modular_group#Relationship_to_hyperbolic_geometry) which is certainly one of the principal sources of interest.2011-12-23
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    There's [Thurston's geometrization conjecture](http://en.wikipedia.org/wiki/William_Thurston#The_geometrization_conjecture).2011-12-23
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    Uniformization (http://en.wikipedia.org/wiki/Uniformization_theorem) is among the most important results in the theory of Riemann surfaces2011-12-23
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    Again, I'm not the right person to give a full-fledged answer here, but I would like to make the case for perhaps looking at the question in a slightly different way. From my point of view, hyperbolic geometry isn't an "idea that has applications" as much as a phenomenon that pops up throughout mathematics. I would classify both of the results already mentioned (geometrization conjecture and the uniformization theorem) as examples of hyperbolic geometry as a *phenomenon* (i.e. the idea that lots of manifolds are naturally hyperbolic) rather than examples of applications.2011-12-23
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    One thing I'll throw out there while waiting for someone who knows more about this than I do is that the field of geometric group theory is all about studying the ways in which groups can be assigned a geometry, which is often hyperbolic. The geometry of the group has algebraic consequences; for instance [hyperbolic groups](http://en.wikipedia.org/wiki/Hyperbolic_group) have solvable word problem.2011-12-23
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    I wrote a comment that disappeared so I'll write it again and hope it doesn't show up twice. Marilyn vos Savant wrote a book claiming Wiles' proof of Fermat's Last Theorem was wrong because it relied on hyperbolic geometry. She was led to this (absurd) conclusion by Wiles' use of modular forms, as referenced by t.b. a few comments up.2011-12-24
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    I studied some spherical geometry when I was in highschool. It was a great tool for studying astronomical events. basiclly in order to undrestand how the sun moves from a person living on earth, you have to know spherical geometry.2016-08-20

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