Classifying GL(2,Z) Z²-orbits by subgroups of R

الأصل · EN

Let G₂ denote the affine group GL(2,Z) Z². For every point x=(x₁,x₂) ∈ 2 let (x)={y∈2 y=γ(x) for some γ∈ G₂ }. Let Gₓ be the subgroup of the additive group R generated by x₁,x₂, 1. If (Gₓ)∈ {1,3} then (x)={y∈2 Gy=Gₓ}. If (Gₓ)=2, knowledge of Gₓ is not sufficient in general to uniquely recover (x): rather, Gₓ classifies precisely (1,ϕ(d)/2) different orbits, where d is the denominator of the smallest positive nonzero rational in Gₓ and ϕ is Euler function. To get a complete classification, polyhedral geometry provides an integer cₓ≥ 1 such that (y)=(x) iff (Gₓ,cₓ)=(Gy,cy).