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

Abstract

A classical result of J\o rgensen 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, 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, we outline this work of Fujiwara and Sela and discuss related results.