An Uncountable Family of Non Orbit Equivalent Actions of $\Bbb F_n$

Abstract

For each $2 \leq n \leq \infty$, we construct an uncountable family of free ergodic measure preserving actions $\alpha_t$ of the free group $\Bbb F_n$ on the standard probability space $(X, \mu)$ such that any two are non orbit equivalent (in fact, not even stably orbit equivalent). These actions are all ``rigid'' (in the sense of [Po01]), with the II$_1$ factors $L^\infty(X, \mu)\rtimes_{\alpha_t} \Bbb F_n$ mutually non stably isomorphic (even non-stably isomorphic) and in the class $\Cal H\Cal T_{_{s}}.$