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

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

2003-10-25

arxiv.org/abs/math/0310400v3

Mathematics

Operator Algebras

11 pages; reorganized and simplified version

Scientific paper

We classify polycyclic totally ordered dimension groups, i.e. dimension groups generated by dense embeddings of the lattice Z^n in the real line R. Our method is based on geometry of simple geodesics on the modular surface of genus g>1. The main theorem says that isomorphism classes of the polycyclic totally ordered dimension group are bijective with the reals modulo the action of the group GL(2,Z). The result is an extension of the Effros-Shen classification of the dicyclic dimension groups.

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