Random walks, WPD actions, and the Cremona group

We study random walks on the Cremona group. We show that almost surely the dynamical degree of a sequence of random Cremona transformations grows exponentially fast, and a random walk produces infinitely many different normal subgroups with probability 1. Moreover, we study the structure of such random subgroups. We prove these results in general for groups of isometries of (non-proper) hyperbolic spaces which possess at least one WPD element. As another application, we answer a question of Margalit showing that a random normal subgroup of the mapping class group is free.