Zd-odometers and cohomology

Abstract

Cohomology for actions of free abelian groups on the Cantor set has (when endowed with an order structure) provided a complete invariant for orbit equivalence. In this paper, we study a particular class of actions of such groups called odometers (or profinite actions) and investigate their cohomology. We show that for a free, minimal Zd-odometer, the first cohomology group provides a complete invariant for the action up to conjugacy. This is in contrast with the situation for orbit equivalence where it is the cohomology in dimension d which provides the invariant. We also consider classification up to isomorphism and continuous orbit equivalence.