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

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Dongryul Kim (Yale)

Abstract: 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. 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.