Appears in collection : Bourbaki - Avril 2023

A classical result of Jørgensen and Thurston shows that the set of volumes of finite volume complete hyperbolic $3$-manifolds is a well-ordered subset of the real numbers of order type $\omega^\omega$; moreover, they showed that each volume can only be attained by finitely many isometry types of hyperbolic $3$-manifolds.

Fujiwara and Sela established a group-theoretic companion of this result: If $\Gamma$ is a non-elementary hyperbolic group, then the set of exponential growth rates of $\Gamma$ is well-ordered, the order type is at least $\omega^\omega$, and each growth rate can only be attained by finitely many finite generating sets (up to automorphisms).

In this talk, I will outline this work of Fujiwara and Sela and discuss related results.

[After Koji Fujiwara and Zlil Sela]