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arXiv · 2509.26444

$\mathbb{Z}^{2}$-dimension groups

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

We study a class of simple dimension groups in which the cyclic subgroup generated by the order unit is replaced by a copy of $\mathbb{Z}^{2}$ satisfying some strict conditions. Our main results are necessary and sufficient conditions on a Bratteli diagram which provides inductive limit structures for such groups. This result has an important application in constructing a version of the Bratteli-Vershik model for minimal actions of $\mathbb{Z}^{2}$ on the Cantor set which will be the subject of a subsequent paper.

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Thierry Giordano, Ian F. Putnam, Christian F. Skau. 2025-09-30. $\mathbb{Z}^{2}$-dimension groups. https://arxiv.org/abs/2509.26444

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