Quantization of Anosov foliations and noncommutative geometry

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The purpose of this project is to obtain a quantum version of the results in "The Godbillon-Vey cyclic cocycle and longitudinal Dirac operators (with the investigator Hitoshi Moriyoshi)" and "Topological approach to quantum surfaces( with Ryszard Nest of the University of Copenhagen)", more precisely to construct noncommutative Anosov foliations on "the unit tangent bundles" over noncommutative Riemann surfaces. This noncommutative Anosov foliations are regarded as quantizations of the (commutative) Anosov foliations associated with geodesic flows on the unit tangent bundles. The ultimate goal of the project is to prove the foliation index theorem of A. Connes, for the noncommutative Anosov foliations.In a joint project with Nest (unpublished) we constructed noncommutative 3-manifolds as strict quantizations of unit circle bundles of closed Riemann surfaces of genus greater than 1.These noncommutaive 3-manifolds were constructed in such a way that the relationship between the Riemann surface and its unit tangent bundle is kept intact through a suitable group action. Moreover we constructed a foliation on the noncommutaive 3-manifold as a certain C^*-algebra in the spirit of A. Connes's noncommutative geometry. We are preparing a paper "Noncommutaive Anosov foliations (tentative title)". We are currently working on detail. As one expects, on view of commutative case, the C^*-algebra representing a "leaf of the noncommutaive foliation is a covering space. We developed some idea how to lift the Dirac operator on the quantized Riemann surface to a longitudinal elliptic operator for the noncommutative Anosov foliation.Unfortunately we were unable to complete the project. However, we certainly continue to work on the project, as we now have a clear idea how to achieve the goal.