K-invariants of conjugacy classes of pseudo-Anosov diffeomorphisms and hyperbolic 3-manifolds
نویسنده
چکیده
New invariants of 3-dimensional manifolds appearing in the Ktheory of certain operator algebras are introduced. First, we consider the conjugacy problem for pseudo-Anosov diffeomorphisms of a compact surface X. The operator algebra in question is an AF -algebra attached to stable (unstable) foliation of the pseudo-Anosov diffeomorphism. We prove that conjugacy classes of commensurable pseudoAnosov diffeomorphisms are bijective with triples (Λ, i, [I]) consisting of order Λ in an algebraic number field K, embedding i : K → R and equivalence class of ideals [I] in Λ. As a consequence, one gets numerical invariants of such classes: determinant ∆ and signature σ which we compute for the case of Anosov diffeomorphisms. Our approach is a blend of geometric topology and K-theory of operator algebras.
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