Geometric covering arguments and ergodic theorems for free groups

Abstract

We present a new approach to the proof of ergodic theorems for actions of non-amenable groups, and give here a complete self-contained account of it in the case of free groups. Our approach is based on direct geometric covering arguments and asymptotic invariance arguments generalizing those developed in the ergodic theory of amenable groups. The results we describe go beyond those previously established for measure-preserving actions of free groups, and demonstrate the significant role the boundary action of the free group plays in the ergodic theory of its measure-preserving actions. Furthermore, our approach suggests the possibility of putting the ergodic theory of amenable groups and nonamenable groups on an equal footing : both can be viewed as special cases in the general ergodic theory of amenable ergodic equivalence relations with finite invariant measure.