Classifying orbits of the affine group over the integers

For each n = 1; 2, . . . , let GL(n, Z) (sic) Z(n) be the affine group over the integers. For every point x = (x(1), . . . ,x(n)) is an element of R-n let orb(x) = {gamma(x) is an element of R-n vertical bar gamma is an element of GL(n, Z) (sic) Z(n)}. Let G(x) be the subgroup of the additive group R generated by x(1), . . . ,x(n), 1. If rank. (G(x)) not equal n then orb(x) = {gamma is an element of R-n vertical bar G(gamma) =G(x)}. Thus, G(x) is a complete classifier of orb(x). By contrast, if rank(G(x)) = n, knowledge of G(x) alone is not sufficient in general to uniquely recover orb(x); as a matter of fact, G(x) determines precisely max(1, phi(d)/2) different orbits, where d is the denominator of the smallest positive non-zero rational in G(x) and phi is the Euler function. To get a complete classification, rational polyhedral geometry provides an integer 1 <= c(x) <= max(1, d/2) such that orb(y) = orb(x) if and only if (G(x), c(x)) = (G(y), c(y)). Applications are given to lattice-ordered abelian groups with strong unit and to AF C*-algebras.