Hyperbolic geometry, continued fractions and classification of the finitely generated totally ordered dimension groups

Abstract

We classify the polycyclic totally ordered simple dimension groups, i.e. dimension groups given by a dense embedding of n-dimensional lattice into the real line. Our method is based on the geometry of simple geodesics on the hyperbolic surface of genus greater or equal two. The main theorem says that isomorphism classes of the polycyclic totally ordered dimension groups are bijective with a generic subset of reals modulo the action of group GL(2,Z). The result is an extension of the Effros-Shen classification of the dicyclic dimension groups.