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 w^w; 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 w^w, 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.
