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

Dongryul Kim (Yale)

Abstract: In this talk, we discuss random walks on isometry groups of Gromov hyperbolic spaces and Teichm├╝ller 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. We also 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.