Arithmetic of hyperbolic 3-manifolds

Abstract

This note is an elaboration of the ideas and intuitions of Grothendieck and Weil concerning the "arithmetic topology". Given 3-dimensional manifold M fibering over the circle we introduce an real quadratic number field K with discriminant d, where d>0 is an integer number uniquely determined by M. The idea is to relate geometry of M to the arithmetic of field K. On this way, we show that Vol M is a limit density of ideals of given norm in the field K (Dirichlet density). The second statement says that the number of cusp points of manifold M is equal to the class number of the field K. It is remarkable that both of the invariants can be explicitly calculated for the concrete values of discriminant d. Our approach is based on the K-theory of noncommutative C*-algebras coming from measured foliations and geodesic laminations studied by Thurston et al. We apply the elaborated technique to solve the Poincare conjecture for given class of manifolds.