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In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely :84(''g'' − 1) automorphisms, where ''g'' is the genus of the surface. This number is maximal by virtue of Hurwitz's theorem on automorphisms . They are also referred to as Hurwitz curves, interpreting them as complex algebraic curves (complex dimension 1 = real dimension 2). The Fuchsian group of a Hurwitz surface is a finite index torsionfree normal subgroup of the (ordinary) (2,3,7) triangle group. The finite quotient group is precisely the automorphism group. Automorphisms of complex algebraic curves are ''orientation-preserving'' automorphisms of the underlying real surface; if one allows orientation-''reversing'' isometries, this yields a group twice as large, of order 168(''g'' − 1), which is sometimes of interest. A note on terminology – in this and other contexts, the "(2,3,7) triangle group" most often refers, not to the ''full'' triangle group Δ(2,3,7) (the Coxeter group with Schwarz triangle (2,3,7) or a realization as a hyperbolic reflection group), but rather to the ''ordinary'' triangle group (the von Dyck group) ''D''(2,3,7) of orientation-preserving maps (the rotation group), which is index 2. The group of complex automorphisms is a quotient of the ''ordinary'' (orientation-preserving) triangle group, while the group of (possibly orientation-reversing) isometries is a quotient of the ''full'' triangle group. == Examples == The Hurwitz surface of least genus is the Klein quartic of genus 3, with automorphism group the projective special linear group PSL(2,7), of order 84(3−1) = 168 = 22·3·7, which is a simple group; (or order 336 if one allows orientation-reversing isometries). The next possible genus is 7, possessed by the Macbeath surface, with automorphism group PSL(2,8), which is the simple group of order 84(7−1) = 504 = 22·32·7; if one includes orientation-reversing isometries, the group is of order 1,008. An interesting phenomenon occurs in the next possible genus, namely 14. Here there is a triple of distinct Riemann surfaces with the identical automorphism group (of order 84(14−1) = 1092 = 22·3·7·13). The explanation for this phenomenon is arithmetic. Namely, in the ring of integers of the appropriate number field, the rational prime 13 splits as a product of three distinct prime ideals. The principal congruence subgroups defined by the triplet of primes produce Fuchsian groups corresponding to the first Hurwitz triplet. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Hurwitz surface」の詳細全文を読む スポンサード リンク
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