Probabilistic Tits alternative for circle diffeomorphisms

TL;DR

This paper proves that under certain probabilistic conditions, two independent random walks on groups of circle diffeomorphisms almost surely generate a free group after some finite steps.

Contribution

It establishes a probabilistic version of Tits' alternative for groups of circle diffeomorphisms, extending previous results to a random setting with minimal moment conditions.

Findings

Almost sure generation of free groups by random walks

Extension of Tits' alternative to probabilistic setting

Applicable to groups with minimal regularity assumptions

Abstract

Let $\mu_1, \mu_2$ be probability measures on $\mathrm{Diff}^1_+(S^1)$ satisfying a suitable moment condition and such that their supports genererate discrete groups acting proximally on $S^1$. Let $(f^n_\omega)_{n \in \mathbb{N}}, (f^n_{\omega'})_{n \in \mathbb{N}}$ be two independent realizations of the random walk driven by $\mu_1, \mu_2$ respectively. We show that almost surely there is an $N \in \mathbb{N}$ such that for all $n \geq N$ the elements $f^n_\omega, f^n_{\omega'}$ generate a nonabelian free group. The proof is inspired by the strategy by R. Aoun for linear groups and uses work of A. Gorodetski, V. Kleptsyn and G. Monakov, and of P. Barrientos and D. Malicet. A weaker (and easier) statement holds for measures supported on $\mathrm{Homeo}_+(S^1)$ with no moment conditions.