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    $\begingroup$ The direction of travel is usually (but not always) in the other direction. Namely, there is a cool idea in geometry (geodesics, volumes, curvature, ...) and it gets imported into geometric group theory. Gromov was an early proponent of this; perhaps his most famous work is "Hyperbolic groups". $\endgroup$
    – Sam Nead
    Commented Sep 7, 2023 at 16:31
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    $\begingroup$ Often it is possible to prove pairs of theorems like what you describe by looking for quasi-isometry invariants. This works because the universal cover of a compact Riemannian manifold is quasi-isometric to its fundamental group, and so any quasi-isometrically invariant fact about one will automatically apply to the other. Many results about compactifications take this form. $\endgroup$
    – Paul Siegel
    Commented Sep 8, 2023 at 2:40
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    $\begingroup$ The analogy in the example you cite is more subtle than most. As others have mentioned, many results from Riemannian geometry can be "coarsened" to more general results in geometric group theory -- this is the founding motivation for the whole subject. But the Fujiwara--Sela result really is just somehow analogous to the Jorgensen--Thurston theorem -- there are similarities but also important differences, and neither is more general than the other. $\endgroup$
    – HJRW
    Commented Sep 8, 2023 at 10:20