The University of Michigan Combinatorics Seminar
We describe the Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces) using graphs dual to the one introduced by Fomin, Shapiro and D.Thurston. We discuss the hyperbolic geometrical origin of the new parameters and describe both classical and quantum theories having the proper number of Thurston variables (foliation-shear coordinates), mapping-class group invariance (both classical and quantum), Poisson and quantum algebra of geodesic functions, and classical and quantum braid-group relations. These new algebras can be defined on the double of the corresponding graph related (in a novel way) to a double of the Riemann surface (which is a Riemann surface with holes, not a smooth Riemann surface). We enlarge the mapping class group allowing transformations relating different Teichmüller spaces of bordered surfaces of the same genus, same number of boundary components, and same total number of marked points but with arbitrary distributions of marked points among the boundary components. We describe the classical and quantum algebras and braid group relations for particular sets of geodesic functions corresponding to An and Dn algebras.