An Uncountable Family of Non Orbit Equivalent Actions of ${\mathbb{F}}_n$

Abstract

For each $2\le n\le \infty$, we construct an uncountable family of free ergodic measure preserving actions ${\alpha }_t$ of the free group ${\mathbb{F}}_n$ 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 $L^{\infty }(X,\mu ){⋊}_{{\alpha }_t}{\mathbb{F}}_n$ mutually non stably isomorphic (even non-stably isomorphic) and in the class $\Cal H\Cal T_{_{s}}.$

New version (13 pages), which includes the case $n=\infty$; revised version 04/08/04