Classifying orbits of the affine group over the integers
Classifying orbits of the affine group over the integers
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 …