Abstract
This paper presents an exposition of the analysis via orbifolds of the isometries with isolated fixed points of compact surfaces, giving a view of the classification of the five regular solids, following the treatment by William Thurston. Then it is shown that a closely related method can be used to recover the classical theorem of Hurwitz that the group of holomorphic automorphisms of a compact Riemann surface of genus g>1 has order at most 84(g-1).
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