Classifying $GL(2,\mathbb Z) \ltimes \mathbb Z^{2}$-orbits by subgroups of $\mathbb R$

Abstract: Let $\mathcal G_2$ denote the affine group $GL(2,\mathbb Z) \ltimes \mathbb Z^{2}$. For every point $x=(x_1,x_2) \in \R2$ let $\orb(x)={y\in\R2\mid y=\gamma(x)$ for some $\gamma \in \mathcal{G}2 }$. Let $G{x}$ be the subgroup of the additive group $\mathbb R$ generated by $x_1,x_2, 1$. If $\rank(G_x)\in {1,3}$ then $\orb(x)={y\in\R2\mid G_y=G_x}$. If $\rank(G_x)=2$, knowledge of $G_x$ is not sufficient in general to uniquely recover $\orb(x)$: rather, $G_x$ classifies precisely $\max(1,\phi(d)/2)$ different orbits, where $d$ is the denominator of the smallest positive nonzero rational in $G_x$ and $\phi$ is Euler function. To get a complete classification, polyhedral geometry provides an integer $c_x\geq 1$ such that $\orb(y)=\orb(x) $ iff $(G_{x},c_{x})=(G_{y},c_{{y}})$.