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 -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 which provides the invariant. We also consider classification up to isomorphism and continuous orbit equivalence.
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Thierry Giordano, Ian F. Putnam, Christian F. Skau, -odometers and cohomology. Groups Geom. Dyn. 13 (2019), no. 3, pp. 909–938DOI 10.4171/GGD/509