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

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

2006-05-06

arxiv.org/abs/math/0605157v4

Mathematics

Geometric Topology

17 pages, 1 fig; a talk at the Banach Center, Bedlewo'06

Scientific paper

The conjugacy problem for the pseudo-Anosov automorphisms of a compact surface is studied. To each pseudo-Anosov automorphism f, we assign an AF-algebra A(f) (an operator algebra). It is proved that the assignment is functorial, i.e. every f', conjugate to f, maps to an AF-algebra A(f'), which is stably isomorphic to A(f). 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 (L, [I], K), where L is an order in the ring of integers in a real algebraic number field K and [I] an equivalence class of the ideals in L. The numerical invariants include the determinant D and the signature S, which we compute for the case of the Anosov automorphisms. A question concerning the p-adic invariants of the pseudo-Anosov automorphism is formulated.

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