In the Wikipedia page for Riemannian Geometry, it mentions that the field had made a "profound impact on group theory".

What are some examples of this?

Looking around a bit (including on that page), it seems more like it is the other way around (i.e. group theory informs Riemannian manifold theory). E.g. analyzing a manifold with its fundamental group, or with Lie theory. (But perhaps these can be viewed inversely.)

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    $\begingroup$ See Gromov's theory of hyperbolic groups. $\endgroup$
    – user98602
    Jan 3 '17 at 17:29
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    $\begingroup$ Dehn's formulation of the word problem in general finitely generated groups arose from his solution of that problem in a particular case, namely surface groups, which uses negative curvature properties of the geometry of the hyperbolic plane. $\endgroup$
    – Lee Mosher
    Jan 3 '17 at 17:57

Felix Klein and later Élie Cartan introduced the study of geometric structures on manifolds through algebra (and in particular through group theory). This is a mutual relationship and one may argue that the geometry motivates the study of group theory (or algebra in general), or vice versa, that the study of group theory (or algebra) is applied to geometry. In some cases this relationship is even "a bijection", e.g., compact flat Riemannian manifolds are classified by Bieberbach groups. Crystallographic groups and its generalisations are perhaps an example which were studied originally as symmetry groups of crystals, but later received an "profound impact" from Riemannian (and pseudo-Riemannian) manifolds and its fundamental groups. In a similar way, there is an impact from number theory, by the theory of discrete groups, and lattices of Lie groups, etc.

The generalisations to other geometric structures and therefore for other algebraic structures or number theory is still active and very modern, e.g., infra-nilmanifolds, affine and projective manifolds, see Milnor, and many more examples.


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