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.[-]

The theory of group actions on CAT(0) cube complexes has exerted a strong influence on geometric group theory and low-dimensional topology in the last two decades. Indeed, knowing that a group G acts properly and cocompactly on a CAT(0) cube complex reveals a lot of its algebraic structure. However, in general, "cubulations'' are non-canonical and the group G can act on cube complexes in many different ways. It is thus natural to try and formulate a good notion of "space of all cubulations of G'', which would prove useful in the study of Out(G) for quite general groups G. I will describe some results in this direction, based on joint works with J. Beyrer and M. Hagen.[-]

A spin structure on a closed surface $S$ of genus $g \geq 2$ is a covering of the unit tangent bundle of $S$ witch restricts to a standard covering of the fiber. Such a spin structure has a parity, even or add. The spin mapping class is the stabilizer of such a spin structure in the mapping class group of $S$. We use a subgraph of the curve graph to construct an explicit generating set of the spin mapping class group consisting of Dehn twists about a system of $2g-1$ simple closed curves.[-]