Giulio Tiozzo (University of Toronto)

Title: Random walks and WPD actions

We study groups of delta-hyperbolic spaces which are not necessarily proper. Such examples occur often in geometry and topology, most notably the action of the mapping class group on the curve complex, or of Out(F_n) on the free factor complex. Since the action is not proper, one needs a weak properness condition, namely the WPD condition formulated by Bestvina-Fujiwara. Under this condition, we prove that for random walks on isometry groups of delta-hyperbolic spaces generic elements are WPD, and the normal closure of a generic element is a free group. This answers a question of D. Margalit for the mapping class group, and for the Cremona group gives a new proof of the abundance of normal subgroups due to Cantat-Lamy. Joint with Joseph Maher.