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In [[Riemann surface]] theory and [[hyperbolic geometry]], the '''Macbeath surface''', also called '''Macbeath's curve''' or the '''Fricke–Macbeath curve''', is the genus-7 [[Hurwitz surface]].

The [[automorphism group]] of the Macbeath surface is the [[simple group]] [[Projective linear group|PSL(2,8)]], consisting of 504 symmetries.<ref name="w">{{harvtxt|Wohlfahrt|1985}}.</ref>

==Triangle group construction==

The surface's [[Fuchsian group]] can be constructed as the principal congruence subgroup of the [[(2,3,7) triangle group]] in a suitable tower of principal congruence subgroups. Here the choices of quaternion algebra and [[Hurwitz quaternion order]] are described at the triangle group page. Choosing the ideal <math>\langle 2 \rangle</math> in the ring of integers, the corresponding principal congruence subgroup defines this surface of genus 7. Its [[systolic geometry|systole]] is about 5.796, and the number of systolic loops is 126 according to R. Vogeler's calculations.

==Historical note==

This surface was originally discovered by {{harvs|first=Robert|last=Fricke|authorlink=Robert Fricke|year=1899|txt}}, but named after [[Alexander Macbeath|Alexander Murray Macbeath]] due to his later independent rediscovery of the same curve.<ref>{{harvtxt|Macbeath|1965}}.</ref> Elkies writes that the equivalence between the curves studied by Fricke and Macbeath "may first have been observed by [[Jean-Pierre Serre|Serre]] in a 24.vii.1990 letter to [[Shreeram Shankar Abhyankar|Abhyankar]]".<ref>{{harvtxt|Elkies|1998}}.</ref>

==See also==
* [[Klein quartic]]
* [[First Hurwitz triplet]]

==Notes==
{{reflist}}

==References==
{{refbegin}}
*{{citation|last1=Berry|first1=Kevin|last2=Tretkoff|first2=Marvin|contribution=The period matrix of Macbeath's curve of genus seven|title=Curves, Jacobians, and abelian varieties, Amherst, MA, 1990|pages=31–40|publisher=Contemp. Math., 136, Amer. Math. Soc.|location=Providence, RI|year=1992|mr=1188192 }}.
*{{citation|last1=Bujalance|first1=Emilio|last2=Costa|first2=Antonio F.|contribution=Study of the symmetries of the Macbeath surface|title=Mathematical contributions|pages=375–385|publisher=Editorial Complutense|location=Madrid|year=1994|mr=1303808 }}.
*{{citation|last=Elkies|first=N. D.|authorlink=Noam Elkies|contribution=Shimura curve computations|title=Algorithmic Number Theory: Third International Symposium, ANTS-III|publisher=Springer-Verlag, Lecture Notes in Computer Science 1423|year=1998|doi=10.1007/BFb0054849|arxiv=math.NT/0005160 |pages=1–47|volume=1423|series=Lecture Notes in Computer Science|editor1-last=Buhler|editor1-first=Joe P.|isbn=3-540-64657-4}}.
*{{citation|last=Fricke|first=R.|authorlink=Robert Fricke|title=Ueber eine einfache Gruppe von 504 Operationen|journal=[[Mathematische Annalen]]|volume=52|year=1899|pages=321–339|doi=10.1007/BF01476163|issue=2–3}}.
*{{citation|last=Gofmann|first=R.|title=Weierstrass points on Macbeath's curve|journal=Vestnik Moskov. Univ. Ser. I Mat. Mekh.|year=1989|issue=5|pages=31–33|volume=104|mr=1029778 }}. Translation in ''Moscow Univ. Math. Bull.'' '''44''' (1989), no. 5, 37–40.
*{{citation|last=Macbeath|first=A.|authorlink=Alexander Macbeath|title=On a curve of genus 7|journal=[[Proceedings of the London Mathematical Society]]|volume=15|year=1965|pages=527–542|doi=10.1112/plms/s3-15.1.527}}.
*{{citation|last=Vogeler|first=R.|title=On the geometry of Hurwitz surfaces|journal=Florida State University thesis|year=2003}}.
*{{citation|last=Wohlfahrt|first=K.|title=Macbeath's curve and the modular group|journal=Glasgow Math. J.|volume=27|year=1985|pages=239–247|mr=0819842 |doi=10.1017/S0017089500006212}}. Corrigendum, vol. 28, no. 2, 1986, p. 241, {{MR|0848433}}.
{{refend}}

[[Category:Hyperbolic geometry]]
[[Category:Riemann surfaces]]
[[Category:Riemannian geometry]]
[[Category:Differential geometry of surfaces]]
[[Category:Systolic geometry]]

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