As you can see from my other question I concern mmyself with the following article at the moment:

Koji Fujiwara, Zlil Sela, The rates of growth in a hyperbolic group, Invent. math. 233 (2023) pp 1427–1470, doi:10.1007/s00222-023-01200-w, arXiv:2002.10278.

One main result of the article is the following: The exponential growth rates of a (non-elementary) hyperbolic group with respect to all of its finite generating sets is well ordered, it satisfies a finite ambiguity condition and the ordinal of this well ordered set is at least $\omega^{\omega}$. There is the conjecture that the ordinal even equals $\omega^{\omega}$.

There also exists the following article:

Gromov, M. Hyperbolic manifolds according to Thurston and Jorgensen. Seminaire Bourbaki, Vol. 1979/80, Lecture Notes in Mathematics, Vol. 842, S. 40-53, Springer, Berlin/Heidelberg 2006, doi: https://doi.org/10.1007/BFb0089927

The result of this article is the folowing: The set of volumes of finite volumed hyperbolic 3-manifolds is well-ordered, satisfies a finite ambiguity condition and the corresponding growth rate is $\omega^{\omega}$.

As you can see, these theorems a pretty similar. (Also the proofs are very similar.)

My question is the following: Do you know some other result from geometric group theory that has a "copy" in the world of Riemannian geometry? You dont have to do any search effort but maybe you already know such an example.