K-theory of hyperbolic 3-manifolds

The subject of present note are relationships between certain class of noncommutative C∗-algebras and geometry of 3-dimensional manifolds which fiber over the circle. We suggest a new classification of such manifolds which is based on the K-theory of a C∗-algebra coming from measured foliations and geodesic laminations studied by Thurston et al. In the first part of the paper a bijection between surface bundles with the pseudoAnosov monodromy and stationary Bratteli diagrams is established. In the second part, we apply the elaborated calculus (rotation numbers, associated number fields) to estimate the volume and Dehn surgery invariants of the 3-dimensional manifolds. In particular, it is shown that the number of cusps is equal to the class number of a real quadratic field associated to the manifold. This generalizes Bianchi’s formula for the imaginary quadratic fields known since 1892.