Orbifold Riemann surfaces: Teichmüller spaces and algebras of geodesic functions

A fat graph description is given for Teichmüller spaces of\linebreak Riemann surfaces with holes and with ${\mathbb Z}_2$- and ${\mathbb Z}_3$-orbifold points (conical singularities) in the Poincaré uniformization. The corresponding mapping class group transformations are presented, geodesic functions are constructed, and the Poisson structure is introduced. The resulting Poisson algebras are then quantized. In the particular cases of surfaces with $n$ ${\mathbb Z}_2$-orbifold points and with one and two holes, the respective algebras $A_n$ and $D_n$ of geodesic functions (classical and quantum) are obtained. The infinite-dimensional Poisson algebra ${\mathfrak D}_n$, which is the semiclassical limit of the twisted $q$-Yangian algebra $Y'_q(\mathfrak{o}_n)$ for the orthogonal Lie algebra $\mathfrak{o}_n$, is associated with the algebra of geodesic functions on an annulus with $n$ ${\mathbb Z}_2$-orbifold points, and the braid group action on this algebra is found. From this result the braid group actions are constructed on the finite-dimensional reductions of this algebra: the $p$-level reduction and the algebra $D_n$. The central elements for these reductions are found. Also, the algebra ${\mathfrak D}_n$ is interpreted as the Poisson algebra of monodromy data of a Frobenius manifold in the vicinity of a non-semisimple point.
Bibliography: 36 titles.