Graph description of Teichmuller spaces and algebras of geodesic functions

Since pioneering works of Penner, the fat graph (combinatorial) description of moduli spaces of Riemann surfaces with punctures advanced far enough to incorporate cases of Riemann surfaces with holes (V.V.Fock) and recently Riemamm surfaces with orbifold points (Fock+Goncharov, L.Ch.). Probably the most interesting object are algebras of geodesic functions (governed mainly by the Goldman bracket) that appear in this approach. I present, first, a short excursion into the graph description of moduli spaces, introduce general algebras of geodesic functions (classical and quantum) and then specify these algebras to two important cases related to A_n and D_n algebras and to algebras of Stokes parameters (Dubrovin, Ugaglia) and to algebras of the groupoid of upper triangular matrices (Bondal).