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Leonid Chekhov

Commutative and noncommutative Teichmuller spaces via graphs

We use the graph description of coordinates on Teichmuller spaces of Riemann surfaces with holes due to Penner and Fock. We explicitly construct the set of invariant objects (lengths of closed geodesics) and the mapping class as well as the modular transformations in terms of these coordinates. The Goldman's Poisson structure can be recovered; its quantum deformation enables us to introduce noncommutative Teichmuller spaces and to construct the quantum mapping class group and quantum modular transformations. Using the geodesic spectral description, we can find a cell decomposition of Teichmuller (and moduli) spaces in the new coordinates. The relation to the new sphericity theorem by Penner and Sullivan is discussed.