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

Abstract

The conjugacy problem for the pseudo-Anosov automorphisms of a compact surface is studied. To each pseudo-Anosov automorphism φ, we assign an AF C*-algebra Aφ (an operator algebra). It is proved that the assignment is functorial, i.e., every φ0, conjugate to φ, maps to an AF C*-algebra Aφ0 , 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 C-algebras. Namely, the main invariant is a triple Δ,[I], K), where 3 is an order in the ring of integers in a real algebraic number field K and [I], an equivalence class of the ideals in δ. The numerical invariants include the determinant 1 and the signature ε, which we compute for the case of the Anosov automorphisms. A question concerning the p-adic invariants of the pseudo-Anosov automorphism is formulated. © 2013 Mathematical Sciences Publishers.