Linear growth of translation lengths of random isometries on Gromov hyperbolic spaces and Teichm\"uller spaces

TL;DR

This paper proves that random walks on isometry groups of Gromov hyperbolic spaces and Teichmüller spaces exhibit at least linear growth in translation lengths, with detailed results on their asymptotic behavior and implications for mapping class groups.

Contribution

It establishes linear growth of translation lengths for non-elementary random walks without moment conditions and extends spectral theorems to Teichmüller spaces under finite first moment.

Findings

Non-elementary random walks have at least linear translation length growth.

Almost all random walks on mapping class groups become pseudo-Anosov.

Almost all random walks on Out(F_n) become fully irreducible.

Abstract

We investigate the translation lengths of group elements that arise in random walks on the isometry groups of Gromov hyperbolic spaces. In particular, without any moment condition, we prove that non-elementary random walks exhibit at least linear growth of translation lengths. As a corollary, almost every random walk on mapping class groups eventually becomes pseudo-Anosov and almost every random walk on $\mathrm{Out}(F_n)$ eventually becomes fully irreducible. If the underlying measure further has finite first moment, then the growth rate of translation lengths is equal to the drift, the escape rate of the random walk. We then apply our technique to investigate the random walks induced by the action of mapping class groups on Teichm{\"u}ller spaces. In particular, we prove the spectral theorem under finite first moment condition, generalizing a result of Dahmani and Horbez.