Appears in collection : Aspects of Non-Positive and Negative Curvature in Group Theory / Courbure négative et courbure négative ou nulle en théorie des groupes

Let $G$ be a hyperbolic group. Its boundary is a topological invariant within the quasi-isometry class of $G$ but it is far from being a complete invariant, e.g. a random group at density ¡1/2 is hyperbolic (Gromov) and its boundary is homeomorphic to the Menger curve (Dahmani-Guirardel-Przytycki) but Mackay proved that there are infinitely many quasi-isometry classes of random groups at density d for small enough d. We discuss the conformal dimension of a hyperbolic group, a quasi-isometry invariant introduced by Pansu. Paulin proved that this is a complete $QI$ invariant of the group. We discuss a technique of Pansu and Bourdon for bounding the conformal dimension from below. We then relate this technique to the family of hyperbolic free by cyclic groups. This is work in progress towards the ultimate goal of showing that there are infinitely many $QI$ classes of free by cyclic groups. This is joint work with Bestvina, Hilion and Stark