# Impact of Riemannian Geometry 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.)

• See Gromov's theory of hyperbolic groups.
– user98602
Jan 3 '17 at 17:29
• 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. Jan 3 '17 at 17:57