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| In [[mathematics]], '''Hurwitz's automorphisms theorem''' bounds the order of the group of [[automorphism]]s, via [[orientation-preserving]] [[conformal mapping]]s, of a compact [[Riemann surface]] of [[genus (mathematics)|genus]] ''g'' > 1, stating that the number of such automorphisms cannot exceed 84(''g''−1). A group for which the maximum is achieved is called a '''Hurwitz group''', and the corresponding Riemann surface a '''[[Hurwitz surface]]'''. Because compact Riemann surfaces are synonymous with non-singular [[algebraic curve|complex projective algebraic curves]], a Hurwitz surface can also be called a '''Hurwitz curve'''.<ref>Technically speaking, there is an [[equivalence of categories]] between the category of compact Riemann surfaces with the orientation-preserving conformal maps and the category of non-singular complex projective algebraic curves with the algebraic morphisms.</ref> The theorem is named after [[Adolf Hurwitz]], who proved it in {{Harv|Hurwitz|1893}}.
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| == Interpretation in terms of hyperbolicity ==
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| One of the fundamental themes in [[differential geometry]] is a trichotomy between the [[Riemannian manifold]]s of positive, zero, and negative [[scalar curvature|curvature]] ''K''. It manifests itself in many diverse situations and on several levels. In the context of compact Riemann surfaces ''X'', via the Riemann [[uniformization theorem]], this can be seen as a distinction between the surfaces of different topologies:
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| * ''X'' a [[Riemann sphere|sphere]], a compact Riemann surface of [[genus (topology)|genus]] zero with ''K'' > 0;
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| * ''X'' a flat [[torus]], or an [[elliptic curve]], a Riemann surface of genus one with ''K'' = 0;
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| * and ''X'' a [[Riemann surface#Hyperbolic Riemann surfaces|hyperbolic surface]], which has genus greater than one and ''K'' < 0.
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| While in the first two cases the surface ''X'' admits infinitely many conformal automorphisms (in fact, the conformal [[automorphism group]] is a complex [[Lie group]] of dimension three for a sphere and of dimension one for a torus), a hyperbolic Riemann surface only admits a discrete set of automorphisms. Hurwitz's theorem claims that in fact more is true: it provides a uniform bound on the order of the automorphism group as a function of the genus and characterizes those Riemann surfaces for which the bound is [[Mathematical jargon#sharp|sharp]].
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| == The idea of a proof and construction of the Hurwitz surfaces ==
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| By the uniformization theorem, any hyperbolic surface ''X'' – i.e., the Gaussian curvature of ''X'' is equal to negative one at every point – is [[covering space|covered]] by the [[Hyperbolic space|hyperbolic plane]]. The conformal mappings of the surface correspond to orientation-preserving automorphisms of the hyperbolic plane. By the [[Gauss-Bonnet theorem]], the area of the surface is
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| : A(''X'') = − 2π χ(''X'') = 4π(''g'' − 1).
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| In order to make the automorphism group ''G'' of ''X'' as large as possible, we want the area of its [[fundamental domain]] ''D'' for this action to be as small as possible. If the fundamental domain is a triangle with the vertex angles π/p, π/q and π/r, defining a [[tessellation|tiling]] of the hyperbolic plane, then ''p'', ''q'', and ''r'' are integers greater than one, and the area is
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|
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| : A(''D'') = π(1 − 1/''p'' − 1/''q'' − 1/''r''). | |
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| Thus we are asking for integers which make the expression
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| :1 − 1/''p'' − 1/''q'' − 1/''r''
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| strictly positive and as small as possible. A remarkable fact is that this minimal value is 1/42, and
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| :1 − 1/2 − 1/3 − 1/7 = 1/42
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| gives a unique (up to permutation) triple of such integers. This would indicate that the order |''G''| of the automorphism group is bounded by
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| : A(''X'')/A(''D'') ≤ 168(''g'' − 1).
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| However, a more delicate reasoning shows that this is an overestimate by the factor of two, because the group ''G'' can contain orientation-reversing transformations. For the orientation-preserving conformal automorphisms the bound is 84(''g'' − 1).
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| === Construction ===
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| [[File:Order-3 heptakis heptagonal tiling.png|thumb|Hurwitz groups and surfaces are constructed based on the tiling of the hyperbolic plane by the (2,3,7) [[Schwarz triangle]].]]
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| To obtain an example of a Hurwitz group, let us start with a (2,3,7)-tiling of the hyperbolic plane. Its full symmetry group is the full [[(2,3,7) triangle group]] generated by the reflections across the sides of a single fundamental triangle with the angles π/2, π/3 and π/7. Since a reflection flips the triangle and changes the orientation, we can join the triangles in pairs and obtain an orientation-preserving tiling polygon.
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| A Hurwitz surface is obtained by 'closing up' a part of this infinite tiling of the hyperbolic plane to a compact Riemann surface of genus ''g''. This will necessarily involve exactly 84(''g'' − 1) double triangle tiles.
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| The following two [[regular tiling]]s have the desired symmetry group; the rotational group corresponds to rotation about an edge, a vertex, and a face, while the full symmetry group would also include a reflection. Note that the polygons in the tiling are not fundamental domains – the tiling by (2,3,7) triangles refines both of these and is not regular.
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| {| class="wikitable"
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| |[[File:Uniform tiling 73-t0.png|100px]]<br>[[order-3 heptagonal tiling]]
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| |[[File:Uniform tiling 73-t2.png|100px]]<br>[[order-7 triangular tiling]]
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| |}
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| [[Wythoff construction]]s yields further [[uniform tiling]]s, yielding [[Order-3_heptagonal_tiling#Wythoff_constructions_from_heptagonal_and_triangular_tilings|eight uniform tilings]], including the two regular ones given here. These all descend to Hurwitz surfaces, yielding tilings of the surfaces (triangulation, tiling by heptagons, etc.).
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| From the arguments above it can be inferred that a Hurwitz group ''G'' is characterized by the property that it is a finite quotient of the group with two generators ''a'' and ''b'' and three relations
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| :<math>a^2 = b^3 = (ab)^7 = 1,\,</math>
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| thus ''G'' is a finite group generated by two elements of orders two and three, whose product is of order seven. More precisely, any Hurwitz surface, that is, a hyperbolic surface that realizes the maximum order of the automorphism group for the surfaces of a given genus, can be obtained by the construction given.
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| This is the last part of the theorem of Hurwitz.
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| == Examples of Hurwitz's groups and surfaces ==
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| [[File:Small cubicuboctahedron.png|thumb|The [[small cubicuboctahedron]] is a polyhedral immersion of the tiling of the [[Klein quartic]] by 20 triangles, meeting at 24 vertices.<ref>{{Harv|Richter}} Note each face in the polyhedron consist of multiple faces in the tiling – two triangular faces constitute a square face and so forth, as per [http://homepages.wmich.edu/~drichter/images/mathieu/hypercolors.jpg this explanatory image].</ref>]]
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| The smallest Hurwitz group is the projective special linear group '''[[PSL(2,7)]]''', of order 168, and the corresponding curve is the [[Klein quartic|Klein quartic curve]]. This group is also isomorphic to '''[[PSL(2,7)|PSL(3,2)]]'''.
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| Next is the [[Macbeath surface|Macbeath curve]], with automorphism group '''PSL(2,8)''' of order 504. Many more finite simple groups are Hurwitz groups; for instance all but 64 of the [[alternating group]]s are Hurwitz groups, the largest non-Hurwitz example being of degree 167. The smallest alternating group that is a Hurwitz group is A<sub>15</sub>.
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| Most [[projective special linear group]]s of large rank are Hurwitz groups, {{harv|Lucchini|Tamburini|Wilson|2000}}. For lower ranks, fewer such groups are Hurwitz. For ''n''<sub>''p''</sub> the order of ''p'' modulo 7, one has that PSL(2,''q'') is Hurwitz if and only if either ''q''=7 or ''q'' = ''p''<sup>''n''<sub>''p''</sub></sup>. Indeed, PSL(3,''q'') is Hurwitz if and only if ''q'' = 2, PSL(4,''q'') is never Hurwitz, and PSL(5,''q'') is Hurwitz if and only if ''q'' = 7<sup>4</sup> or ''q'' = ''p''<sup>''n''<sub>''p''</sub></sup>, {{harv|Tamburini|Vsemirnov|2006}}.
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| Similarly, many [[group of Lie type|groups of Lie type]] are Hurwitz. The finite [[classical group]]s of large rank are Hurwitz, {{harv|Lucchini|Tamburini|1999}}. The [[exceptional Lie group]]s of type G2 and the [[Ree group]]s of type 2G2 are nearly always Hurwitz, {{harv|Malle|1990}}. Other families of exceptional and twisted Lie groups of low rank are shown to be Hurwitz in {{harv|Malle|1995}}.
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| There are 12 [[sporadic groups]] that can be generated as Hurwitz groups: the [[Janko group]]s J<sub>1</sub>, J<sub>2</sub> and J<sub>4</sub>, the [[Fischer group]]s Fi<sub>22</sub> and Fi'<sub>24</sub>, the [[Rudvalis group]], the [[Held group]], the [[Thompson group (finite)|Thompson group]], the [[Harada–Norton group]],the third [[Conway group]] Co<sub>3</sub>, the [[Lyons group]], and the [[Monster group|Monster]], {{harv|Wilson|2001}}.
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| ==See also==
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| *[[(2,3,7) triangle group]]
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| == Notes ==
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| <references/>
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| ==References==
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| {{refbegin}}
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| *{{Citation|last = Hurwitz|first = A.|title = Über algebraische Gebilde mit Eindeutigen Transformationen in sich|journal = [[Mathematische Annalen]]|volume = 41|issue=3 |year = 1893|pages = 403–442|doi = 10.1007/BF01443420 |jfm=24.0380.02|postscript = . }}
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| * {{Citation | last1=Lucchini | first1=A. | last2=Tamburini | first2=M. C. | title=Classical groups of large rank as Hurwitz groups | doi=10.1006/jabr.1999.7911 | mr=1706821 | year=1999 | journal=Journal of Algebra | issn=0021-8693 | volume=219 | issue=2 | pages=531–546}}
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| * {{Citation | last1=Lucchini | first1=A. | last2=Tamburini | first2=M. C. | last3=Wilson | first3=J. S. | title=Hurwitz groups of large rank | doi=10.1112/S0024610799008467 | mr=1745399 | year=2000 | journal=Journal of the London Mathematical Society. Second Series | issn=0024-6107 | volume=61 | issue=1 | pages=81–92}}
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| * {{Citation | doi=10.4153/CMB-1990-059-8 | last1=Malle | first1=Gunter | title=Hurwitz groups and G2(q) | mr=1077110 | year=1990 | journal=[[Canadian Mathematical Bulletin]] | issn=0008-4395 | volume=33 | issue=3 | pages=349–357}}
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| * {{Citation | last1=Malle | first1=Gunter | title=Groups of Lie type and their geometries (Como, 1993) | publisher=[[Cambridge University Press]] | series=London Math. Soc. Lecture Note Ser. | mr=1320522 | year=1995 | volume=207 | chapter=Small rank exceptional Hurwitz groups | pages=173–183}}
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| * {{Citation | last1=Tamburini | first1=M. C. | last2=Vsemirnov | first2=M. | title=Irreducible (2,3,7)-subgroups of PGL(n,F) for n ≤ 7 | doi=10.1016/j.jalgebra.2006.02.030 | mr=2228652 | year=2006 | journal=Journal of Algebra | issn=0021-8693 | volume=300 | issue=1 | pages=339–362}}
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| * {{cite doi|10.1515/jgth.2001.027}}
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| * {{citation | ref = {{harvid|Richter}} | first = David A. | last = Richter | url = http://homepages.wmich.edu/~drichter/mathieu.htm | title = How to Make the Mathieu Group M<sub>24</sub> | accessdate = 2010-04-15 }}
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| {{refend}}
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