Operator algebras and conjugacy problem for the pseudo-Anosov automorphisms of a surface

The conjugacy problem for the pseudo-Anosov automorphisms of a compact surface is studied. To each pseudo-Anosov automorphism φ, we assign an AF -algebra Aφ (an operator algebra). It is proved that the assignment is functorial, i.e. every φ, conjugate to φ, maps to an AF algebra Aφ′ , which is stably isomorphic to Aφ. The new invariants of the conjugacy of the pseudo-Anosov automorphisms are obtained from the known invariants of the stable isomorphisms of the AF -algebras. Namely, the main invariant is a triple (Λ, [I ],K), where Λ is an order in the ring of integers in a real algebraic number field K and [I ] an equivalence class of the ideals in Λ. The numerical invariants include the determinant ∆ and the signature Σ, which we compute for the case of the Anosov automorphisms. A question concerning the p-adic invariants of the pseudo-Anosov automorphism is formulated.