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A classical result of J\o rgensen and Thurston shows that the set of volumesof finite volume complete hyperbolic $3$-manifolds is a well-ordered subset ofthe real numbers of order type~$\omega^\omega$; moreover, each volume can onlybe 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 exponentialgrowth rates of~$\Gamma$ is well-ordered, the order type is atleast~$\omega^\omega$, and each growth rate can only be attained by finitelymany finite generating sets (up to automorphisms).  In this talk, we outline this work of Fujiwara and Sela and discuss relatedresults.

Exponential growth rates in hyperbolic groups (after Koji Fujiwara and Zlil Sela)