This chapter has two main purposes; the first is to classify function groups up to similarity, and regular (i.e., geometrically finite) function groups up to deformation, and the second is to show that every regular covering of a finite Riemann surface, where the covering surface is planar, can be topologically realized by a regular function group. Using similar techniques with quasiconformal mappings, one can prove that every planar regular covering of a finite Riemann surface can be conformally realized by a regular function group; this theorem however is beyond the scope of this book.
KeywordsStructure Loop Function Group Fuchsian Group Simple Loop Schottky Group
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