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

Daniele Mundici

Let ${\mathcal{G}}_2$ denote the affine group $GL(2,\mathbb{Z})⋉{\mathbb{Z}}^2$. For every point $x=(x_1,x_2)\in \mathbb{R}2$ let $\orb(x)=\{y\in\R2\mid y=γ(x)$ for some $γ\in \mathcal…