Leonid Chekhov (Steklov Mathematical Institute)
chekhov@mi.ras.ru We propose a graph description of Teichmüller spaces of surfaces with marked points on boundary components (bordered surfaces). Introducing new parameters, we formulate this theory in terms of hyperbolic geometry. We describe both classical and quantum theories enjoying an (extended) mapping class group invariance and classical and quantum braid-group relations. For particular sets of geodesic functions, we obtain An and Dn algebras known before from the cluster theory side. If we'll have enough time, we'll discuss the relation to the Thurston theory.