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

Abstract

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 $\gamma \in {\mathcal{G}}_2\}$. 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 $\max (1,\varphi (d)/2)$ different orbits, where $d$ is the denominator of the smallest positive nonzero rational in $G_x$ and $\varphi$ is Euler function. To get a complete classification, polyhedral geometry provides an integer $c_x\ge 1$ such that $\orb(y)=\orb(x) $ iff $(G_x,c_x)=(G_y,c_y)$.

Cited by in corpus (1)

• Classifying orbits of the affine group over the integers