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arXiv · 1709.08585

$Z^{d}$-odometers and cohomology

Thierry Giordano·Ian F. Putnam·Christian F. SKau
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Abstract

Cohomology for actions of free abelian groups on the Cantor set has (when endowed with an order structure) provided a complete invariance 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 $\Z^{d}$-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.

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Thierry Giordano, Ian F. Putnam, Christian F. SKau. 2017-09-25. $Z^{d}$-odometers and cohomology. https://arxiv.org/abs/1709.08585

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