Apparaît dans la collection : Geometry of groups and 3-manifolds: state of the art and perspectives / Géométrie des groupes et géométrie des 3-variétés : situation et perspectives

Let $G$ be a torsion-free hyperbolic group, let $S$ be a finite generating set of $G$, and let $f$ be an automorphism of $G$. We want to understand the possible growth types for the word length of $f^n(g)$, where $g$ is an element of $G$. Growth was completely described by Thurston when $G$ is the fundamental group of a hyperbolic surface, and can be understood from Bestvina-Handel's work on train-tracks when $G$ is a free group. We address the general case of a torsion-free hyperbolic group $G$; we show that every element in $G$ has a well-defined exponential growth rate under iteration of $f$, and that only finitely many exponential growth rates arise as $g$ varies in $G$. In addition, we show the following dichotomy: every element of $G$ grows either exponentially fast or polynomially fast under iteration of $f$. This is a joint work with Rémi Coulon, Arnaud Hilion and Gilbert Levitt.