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

Thierry Giordano, Ian F. Putnam, Christian F. Skau

TL;DR

The paper addresses realizing simple dimension groups that embed a dense copy of $\\mathbb{Z}^{2}$ via a map $\\sigma: G \\to \\mathbb{R}^{2}$ as inductive limits of simplicial groups encoded by Bratteli diagrams. It provides a complete characterization of when such inductive-limit data yield a simple dimension group with exactly two extremal states, specifying both the forward construction (sufficiency) and the reverse realization (necessity) through detailed matrix inequalities and cone decompositions, ultimately enabling a Bratteli-Vershik model for minimal $\\mathbb{Z}^{2}$-actions on the Cantor set. The results connect the combinatorics of Bratteli diagrams with the affine structure of the state space, and set the stage for a cohomology-based dynamical model via $H^{1}(X,\\varphi)$. The approach uses a two-component decomposition into cones $G(l,1)$ and $G(l,2)$ and constructive factorization to ensure positivity and density properties, providing an explicit, algorithmic pathway from diagram data to the target dimension group invariant. Collectively, these contributions advance the explicit realization of $\\mathbb{Z}^{2}$-dimension groups and their application to Bratteli-Vershik models of higher-rank actions.

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.

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

TL;DR

The paper addresses realizing simple dimension groups that embed a dense copy of via a map as inductive limits of simplicial groups encoded by Bratteli diagrams. It provides a complete characterization of when such inductive-limit data yield a simple dimension group with exactly two extremal states, specifying both the forward construction (sufficiency) and the reverse realization (necessity) through detailed matrix inequalities and cone decompositions, ultimately enabling a Bratteli-Vershik model for minimal -actions on the Cantor set. The results connect the combinatorics of Bratteli diagrams with the affine structure of the state space, and set the stage for a cohomology-based dynamical model via . The approach uses a two-component decomposition into cones and and constructive factorization to ensure positivity and density properties, providing an explicit, algorithmic pathway from diagram data to the target dimension group invariant. Collectively, these contributions advance the explicit realization of -dimension groups and their application to Bratteli-Vershik models of higher-rank actions.

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 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 on the Cantor set which will be the subject of a subsequent paper.

Paper Structure

This paper contains 3 sections, 13 theorems, 77 equations.

Table of Contents

  1. Introduction and statements of the results
  2. Sufficiency
  3. Necessity

Key Result

Theorem 1.1

Let $G$ be a countable, torsion-free abelian group and let $\sigma:G \rightarrow \mathbb{R}^{2}$ be a group homomorphism satisfying: If we define then $G$ is a simple dimension group. Further, if $u^{1}$ and $u^{2}$ are any elements of $G$ with $\sigma(u^{1}) = (1,0)$ and $\sigma(u^{2}) = (0,1)$, then neither is positive while $u= u^{1} + u^{2}$ is an order unit for $G$. Letting $\sigma_{1}, \si

Theorems & Definitions (22)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Lemma 2.3
  • proof
  • ...and 12 more