Geodesic
Riemann Surfaces with Orbifold Points
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometry, topology, and mathematical physics. II, Tome 266 (2009), pp. 237-262.

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We interpret the previously developed Teichmüller theory of surfaces with marked points on boundary components (bordered surfaces) as the Teichmüller theory of Riemann surfaces with orbifold points of order 2. In the Poincaré uniformization pattern, we describe necessary and sufficient conditions for the group generated by the Fuchsian group of the surface with added inversions to be of the almost hyperbolic Fuchsian type. All the techniques elaborated for the bordered surfaces (quantization, classical and quantum mapping-class group transformations, and Poisson and quantum algebra of geodesic functions) are equally applicable to the surfaces with orbifold points. 
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L. O. Chekhov. Riemann Surfaces with Orbifold Points. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Geometry, topology, and mathematical physics. II, Tome 266 (2009), pp. 237-262. http://geodesic.mathdoc.fr/item/TM_2009_266_a13/

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