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Classifying orbits of the affine group over the integers

Published online by Cambridge University Press:  22 July 2015

LEONARDO MANUEL CABRER  and
DANIELE MUNDICI
LEONARDO MANUEL CABRER
Affiliation:
Department of Statistics, Computer Science and Applications, ‘Giuseppe Parenti’, University of Florence, Viale Morgagni 59 – 50134, Florence, Italy email l.cabrer@disia.unifi.it
DANIELE MUNDICI
Affiliation:
Department of Mathematics and Computer Science ‘Ulisse Dini’, University of Florence, Viale Morgagni 67/A, I-50134 Florence, Italy email mundici@math.unifi.it
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Abstract

For each $n=1,2,\ldots ,$ let $\text{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^{n}$ be the affine group over the integers. For every point $x=(x_{1},\ldots ,x_{n})\in \mathbb{R}^{n}$ let $\text{orb}(x)=\{\unicode[STIX]{x1D6FE}(x)\in \mathbb{R}^{n}\mid \unicode[STIX]{x1D6FE}\in \text{GL}(n,\mathbb{Z})\ltimes \mathbb{Z}^{n}\}.$ Let $G_{x}$ be the subgroup of the additive group $\mathbb{R}$ generated by $x_{1},\ldots ,x_{n},1$. If $\text{rank}(G_{x})\neq n$ then $\text{orb}(x)=\{y\in \mathbb{R}^{n}\mid G_{y}=G_{x}\}$. Thus, $G_{x}$ is a complete classifier of $\text{orb}(x)$. By contrast, if $\text{rank}(G_{x})=n$, knowledge of $G_{x}$ alone is not sufficient in general to uniquely recover $\text{orb}(x)$; as a matter of fact, $G_{x}$ determines precisely $\max (1,\unicode[STIX]{x1D719}(d)/2)$ different orbits, where $d$ is the denominator of the smallest positive non-zero rational in $G_{x}$ and $\unicode[STIX]{x1D719}$ is the Euler function. To get a complete classification, rational polyhedral geometry provides an integer $1\leq c_{x}\leq \max (1,d/2)$ such that $\text{orb}(y)=\text{orb}(x)$ if and only if $(G_{x},c_{x})=(G_{y},c_{y})$. Applications are given to lattice-ordered abelian groups with strong unit and to AF $C^{\ast }$-algebras.

Type
Research Article
Information
Ergodic Theory and Dynamical Systems , Volume 37 , Issue 2 , April 2017 , pp. 440 - 453
Copyright
© Cambridge University Press, 2015 

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