For each $n=1,2,\dots$, let $\mathsf{ GL}(n,\mathbb{ Z})\ltimes \mathbb{ Z}^n$ be the affine group over the integers. For every point $x=(x_1,\dots,x_n) \in \mathbb{ R}^n$ let $\mathrm{ orb}(x)=\{ \gamma(x)\in \mathbb{ R}^n\mid\gamma\in \mathsf{ GL}(n,\mathbb{ Z})\ltimes \mathbb{ Z}^n\}.$ Let $G_{ x}$ be the subgroup of the additive group $\mathbb R$ generated by $x_1,\dots,x_n, 1$. If $\mathrm{ rank}(G_x)\neq n$ 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 $\mathrm{ max}(1,\frac{ \phi(d)}{ 2})$ 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, rational polyhedral geometry provides an integer $1\leq c_x\leq \mathrm{ max}(1,d/2)$ such that $\mathrm{ orb}(y)=\mathrm{ orb}(x) $ iff $(G_{ x},c_{ x})=(G_{ y},c_{ { y}})$.
Categories: math.GR math.MG