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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).

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rdfs:label	Hurwitz surface (en) Поверхность Гурвица (ru) 赫爾維茨曲面 (zh)
rdfs:comment	Поверхность Гурвица — компактная риманова поверхность, имеющая в точности 84(g − 1) автоморфизмов, где g — род поверхности. Их также называют кривыми Гурвица, понимая их при этом как комплексные алгебраические кривые (комплексная размерность 1 соответствует вещественной размерности 2). Названа в честь немецкого математика Адольфа Гурвица. (ru) 在黎曼曲面理論和中，赫爾維茨曲面（英語：Hurwitz surface）是一個緊湊精確的 84(g − 1) 黎曼曲面構造，由阿道夫·赫維茲所命名的曲面。其中g是該曲面的虧格。這個數字是赫維茨對同構定理（Hurwitz 1893）的最大值。若將它們解釋為複數的代數曲線（複數1維=實數2維）的話，也可以將之稱為赫爾維茨曲線。赫爾維茨曲面的是撓的正規子群的有限索引。其有限商群也正好是其同構群。 (zh) 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). (en)
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dcterms:subject	Riemann surfaces Systolic geometry
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Link from a Wikipage to another Wikipage	Prime ideal Projective special linear group Schwarz triangle Macbeath surface Riemann surface Ring of integers Index of a subgroup (2,3,7) triangle group Riemann surfaces Systolic geometry Genus (mathematics) Simple group Mikhail Katz Fuchsian group Mathematische Annalen Adolf Hurwitz Number field Coxeter group Hurwitz quaternion order Hyperbolic geometry Systolic geometry Torsion-free group Reflection group Klein quartic Compact Riemann surface First Hurwitz triplet PSL(2,7) Von Dyck group Principal congruence subgroup Orientation-preserving Hurwitz's theorem on automorphisms
Link from a Wikipage to an external page	http://www.emis.de/journals/BAG/vol.44/no.2/9.html http://www.emis.de/journals/BAG/vol.44/no.2/b44h2sin.pdf http://www.heldermann.de/BAG/bag44.htm http://www.heldermann.de/BAG/bag44.htm%23bag442
sameAs	Hurwitz surface Hurwitz surface Hurwitz surface Hurwitz surface Hurwitz surface Hurwitz surface Hurwitz surface
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has abstract	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. (en) Поверхность Гурвица — компактная риманова поверхность, имеющая в точности 84(g − 1) автоморфизмов, где g — род поверхности. Их также называют кривыми Гурвица, понимая их при этом как комплексные алгебраические кривые (комплексная размерность 1 соответствует вещественной размерности 2). Названа в честь немецкого математика Адольфа Гурвица. (ru) 在黎曼曲面理論和中，赫爾維茨曲面（英語：Hurwitz surface）是一個緊湊精確的 84(g − 1) 黎曼曲面構造，由阿道夫·赫維茲所命名的曲面。其中g是該曲面的虧格。這個數字是赫維茨對同構定理（Hurwitz 1893）的最大值。若將它們解釋為複數的代數曲線（複數1維=實數2維）的話，也可以將之稱為赫爾維茨曲線。赫爾維茨曲面的是撓的正規子群的有限索引。其有限商群也正好是其同構群。 (zh)
gold:hypernym	Surface
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foaf:isPrimaryTopicOf	wikipedia-en:Hurwitz_surface
is Link from a Wikipage to another Wikipage of	List of curves Macbeath surface List of mathematical shapes (2,3,7) triangle group Order-7 triangular tiling Modular group Systoles of surfaces Gallery of curves Adolf Hurwitz Bring's curve Projectively extended real line Hurwitz's automorphisms theorem Hurwitz quaternion order Systolic geometry Heptagonal tiling Klein quartic First Hurwitz triplet Hurwitz curve PSL(2,7)
is Wikipage redirect of	Hurwitz curve
is foaf:primaryTopic of	wikipedia-en:Hurwitz_surface

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