. O A ] 1 7 O ct 2 00 7 Riemann surfaces and AF - algebras

For a generic set in the Teichmüller space, we construct a covariant functor with the range in a category of the operator AF -algebras. The functor maps isomorphic Riemann surfaces to the stably isomorphic AF algebras. Our construction is based on a Hodge theory for measured foliations, elaborated by Hubbard, Masur and Thurston. As an application, we consider the following two problems: (i) a categorical correspondence between complex and noncommutative tori and (ii) faithful representation of the mapping class group of genus g ≥ 2 as a subgroup of the matrix group GL6g−6(Z).