Analogous results in geometric group theory and Riemannian geometry?

Question

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.

Answer 1

I think Cheeger's inequality is a good example.

Riemannian geometry version

Let $M$ be a closed Riemannian $n$-manifold. Say that a $n-1$ dimensional submanifold $N$ separates $M$ if the complement of $N$ has two connected components $A$ and $B$. Define the isoperimetric constant of $M$ to be:

$$\Phi(M) = \inf_N \frac{Vol_{n-1}(N)}{\min(Vol_n(A), Vol_n(B))}$$

where the infimum is taken over all submanifolds that separate $M$.

Cheeger's inequality says that:

$$\Phi(M) \leq 2 \sqrt{\lambda}$$

where $\lambda$ is the smallest positive eigenvalue of the Laplace operator on $M$.

Discrete version

Let $G$ be a connected graph with $n$ vertices. Define the isoperimetric constant of $G$ to be:

$$\Phi(G) = \inf_S \frac{|\partial S|}{\min(|S|, |S^c|)}$$

where $S$ is any set of vertices, and $\partial S$ is the set of all edges that have exactly one endpoint in $S$.

The discrete Cheeger inequality says:

$$\Phi(G) \leq \sqrt{2 \lambda}$$

where $\lambda$ is the smallest nonzero eigenvalue of the discrete Laplace operator for $G$.

(The discrete version is often applied to groups via their Caley graph, and it generalizes to certain kinds of infinite groups.)

Answer 2

Here is a very classical example. As stated in the comments, Gromov was an early proponent of importing ideas from geometry to group theory, but already thirty years earlier there was work in this direction. One of the most fundamental papers on geometric group theory was written in 1955 by A. S. Schwarz [1]. In this paper the famous "Schwarz-Milnor Lemma" is proved; it relates the growth of balls in the universal cover of a compact Riemannian manifold $M$ to the growth of balls in the fundamental group $\pi_1(M)$, and says that they are essentially the same (one speaks of quasi-isometries in modern terminology; this is then the same as the comment made by PaulSiegel).

At the end, he gives some applications of this result. First, he states two results:

Theorem 1. The volume of balls in a simply connected Riemannian manifold whose curvature is $K \leq C$, where $C <0$, cannot grow slower than $e^r$.

Theorem 2. The volume of balls in a simply connected $n$-dimensional Riemannian manifold of non-positive curvature ($K \leq 0$) cannot grow slower than $r^n$.

In both cases, $r$ refers to the radius of the balls in question. He attributes Theorem 2 to follow from results by É. Cartan (1936). Schwarz then uses his main theorem to state the two analogous results for growth in fundamental groups, after observing that the volume of balls in the universal cover of a compact Riemannian manifold with abelian fundamental group of rank $n$ grows as $r^n$:

Theorem 1'. The fundamental group of a Riemannian manifold with negative curvature ($K<0$) cannot be abelian.

Theorem 2'. The fundamental group of an $n$-dimensional compact Riemannian manifold with non-positive curvature $(K \leq 0)$ cannot be abelian of rank $<n$.

Finding the paper is, if I recall right, a bit tricky, even the original Russian one. Albert Schwarz sent me a copy of the Russian, and I translated it into English. I can provide a copy of either, if needed.

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[1] A. S. Schwarz, "The volume invariant of coverings", Dokl. Akad. Nauk SSSR 105:1, 1955.

[Note: sometimes the title of this article is translated as "A volume invariant of coverings". The original Russian title "Объемный инвариант накрывающих" has no article "a/the", as this does not exist in Russian. But one can infer that "the" is meant, as the paper is not about defining some new invariant, but rather about proving facts about an existing invariant; this invariant is attributed in the paper to an earlier article by V. A. Efremovich.]