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

TL;DR

This paper classifies orbits of the affine group acting on the plane using subgroup invariants and polyhedral geometry, revealing how subgroup rank and additional parameters determine orbit structure.

Contribution

It introduces a complete classification of orbits under the affine group using subgroup invariants and polyhedral geometry, extending previous partial results.

Findings

Orbits with subgroup rank 1 or 3 are classified by the subgroup alone.

For rank 2, subgroup alone is insufficient; additional parameters are needed.

Polyhedral geometry provides an integer parameter c_x for full orbit classification.

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 $(G_{x},c_{x})=(G_{y},c_{{y}})$.