SLLN for asymptotic translation lengths of random isometries on Gromov hyperbolic spaces and Teichmüller spaces

In this talk, we discuss random walks on isometry groups of Gromov hyperbolic spaces and Teichmüller spaces. We prove that non-elementary random walks exhibit at least linear growth of asymptotic translation lengths without any moment condition. As a corollary, it follows that almost every sample-path on a mapping class group eventually becomes pseudo-Anosov. Moreover, we show that if the underlying measure has a finite first moment, then the growth is linear, as Strong Law of Large Number. This is joint work with Hyungryul Baik and Inhyeok Choi.