Manchester Geometry Seminar 2005/2006

Based on papers with V.Fock and R.Penner, we propose the way to quantize Teichmüller and Thurston theories for Riemann surfaces with holes (punctures). These surfaces admit, in the Poincare uniformization, a graph description due to Penner and Fock. The corresponding parameters are the coordinates on the Teichmüller space, and the mapping class group (modular) transformations can be explicitly constructed. Introducing the Poincaré structure compatible with the Goldman Poisson brackets for geodesic functions that follow from 2+1-dimensional Chern-Simons theory, we were able to quantize the structure thus producing the quantum mapping class group transformations and quantum geodesic functions. The arising algebras, in some particular cases, are related to Nelson-Regge algebras of geodesics, or to algebras of Stokes parameters in isomonodromic deformations. In the second part of the talk, we consider the Thurston theory of measured geodesic laminations and show that, under the proper definition pertaining to the so-called tropical limit of mapping class group transformations, we can define the (classical and quantum) limits of the geodesic functions for arbitrary measured lamination and prove the existence of these limiting geodesic lengths, or of the corresponding operators describing quantum geodesic lengths.