Random walks, WPD actions, and the Cremona group

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