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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 PSL(2,8), consisting of 504 symmetries.[1]

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 ⟨ 2 ⟩ {\displaystyle \langle 2\rangle } \langle 2\rangle in the ring of integers, the corresponding principal congruence subgroup defines this surface of genus 7. Its 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 Robert Fricke (1899), but named after Alexander Murray Macbeath due to his later independent rediscovery of the same curve.[2] Elkies writes that the equivalence between the curves studied by Fricke and Macbeath "may first have been observed by Serre in a 24.vii.1990 letter to Abhyankar".[3]
See also

Klein quartic
First Hurwitz triplet

Notes

Wohlfahrt (1985).
Macbeath (1965).

Elkies (1998).

References

Berry, Kevin; Tretkoff, Marvin (1992), "The period matrix of Macbeath's curve of genus seven", Curves, Jacobians, and abelian varieties, Amherst, MA, 1990, Providence, RI: Contemp. Math., 136, Amer. Math. Soc., pp. 31–40, doi:10.1090/conm/136/1188192, MR 1188192.
Bujalance, Emilio; Costa, Antonio F. (1994), "Study of the symmetries of the Macbeath surface", Mathematical contributions, Madrid: Editorial Complutense, pp. 375–385, MR 1303808.
Elkies, N. D. (1998), "Shimura curve computations", in Buhler, Joe P. (ed.), Algorithmic Number Theory: Third International Symposium, ANTS-III, Lecture Notes in Computer Science, 1423, Springer-Verlag, Lecture Notes in Computer Science 1423, pp. 1–47, arXiv:math.NT/0005160, doi:10.1007/BFb0054849, ISBN 3-540-64657-4.
Fricke, R. (1899), "Ueber eine einfache Gruppe von 504 Operationen" (PDF), Mathematische Annalen, 52 (2–3): 321–339, doi:10.1007/BF01476163.
Gofmann, R. (1989), "Weierstrass points on Macbeath's curve", Vestnik Moskov. Univ. Ser. I Mat. Mekh., 104 (5): 31–33, MR 1029778. Translation in Moscow Univ. Math. Bull. 44 (1989), no. 5, 37–40.
Macbeath, A. (1965), "On a curve of genus 7", Proceedings of the London Mathematical Society, 15: 527–542, doi:10.1112/plms/s3-15.1.527.
Vogeler, R. (2003), "On the geometry of Hurwitz surfaces", Florida State University thesis.
Wohlfahrt, K. (1985), "Macbeath's curve and the modular group", Glasgow Math. J., 27: 239–247, doi:10.1017/S0017089500006212, MR 0819842. Corrigendum, vol. 28, no. 2, 1986, p. 241, MR0848433.

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