Camille Horbez: Automorphisms of hyperbolic groups and growth

Abstract: 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 fn(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.

Recording during the thematic meeting "Geometry of Groups and 3-manifolds" the February 20, 2018 at the Centre International de Rencontres Mathématiques (Marseille, France)

Filmmaker: Guillaume Hennenfent

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