Hyperbolic geometry, continued fractions and classification of AF C*-algebras

We classify polycyclic dimension groups, i.e. dimension groups with the underlying group Z and n ≥ 4. Our method is based on geometry of simple geodesic lines on the Riemann surface of genus g ≥ 2. The main theorem says that every polycyclic dimension group can be indexed by single real parameter α, where α is a positive irrational modulo the action of GL(2,Z). This result is an extension of the Effros-Shen classification of dicyclic dimension groups ([4]) and has various applications inside and outside C∗-algebra theory.