Orbifold Riemann surfaces and geodesic algebras

We study the Teichm\"uller theory of Riemann surfaces with orbifold points of order two using the fat graph technique. The previously developed technique of quantization, classical and quantum mapping-class group transformations, and Poisson and quantum algebras of geodesic functions is applicable to the surfaces with orbifold points. We describe classical and quantum braid group relations for particular sets of geodesic functions corresponding to $A_n$ and $D_n$ algebras and describe their central elements for the Poisson and quantum algebras.