Classifying orbits of the affine group over the integers

Abstract

For each $n=1,2,\dots$, let $\mathsf{GL}(n,\mathbb{Z})⋉{\mathbb{Z}}^n$ be the affine group over the integers. For every point $x=(x_1,\dots ,x_n)\in {\mathbb{R}}^n$ let $\mathrm{orb}(x)=\{\gamma (x)\in {\mathbb{R}}^n\mid \gamma \in \mathsf{GL}(n,\mathbb{Z})⋉{\mathbb{Z}}^n\}.$ Let $G_x$ be the subgroup of the additive group $\mathbb{R}$ generated by $x_1,\dots ,x_n,1$. If $\mathrm{rank}(G_x)\ne n$ then $\mathrm{orb}(x)=\{y\in {\mathbb{R}}^n\mid G_y=G_x\}$. Thus,$G_x$ is a complete classifier of $\mathrm{orb}(x)$. By contrast, if $\mathrm{rank}(G_x)=n$, knowledge of $G_x$ alone is not sufficient in general to uniquely recover $\mathrm{orb}(x)$: as a matter of fact, $G_x$ determines precisely $\max (1,\frac{\varphi (d)}{2})$ 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, rational polyhedral geometry provides an integer $1\le c_x\le \max (1,d/2)$ such that $\mathrm{orb}(y)=\mathrm{orb}(x)$ iff $(G_x,c_x)=(G_y,c_y)$.

References in corpus (2)

• Approximation to points in the plane by SL(2,Z)-orbits
• Classifying $GL(2,\mathbb{Z})⋉{\mathbb{Z}}^2$-orbits by subgroups of $\mathbb{R}$

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