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 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.[2]
It is possible to realize the resulting triangulated surface as a non-convex polyhedron without self-intersections.[3]
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.[4] 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".[5] In a later survey article Macbeath attributes the result to Fricke.[6]
Berry, Kevin; Tretkoff, Marvin (1992), "The period matrix of Macbeath's curve of genus seven", Curves, Jacobians, and abelian varieties, Amherst, MA, 1990, Contemporary Mathematics, vol. 136, Providence, RI: Contemp. Math., 136, Amer. Math. Soc., pp. 31–40, doi:10.1090/conm/136/1188192, ISBN978-0-8218-5143-2, MR1188192.
Bokowski, Jürgen; Cuntz, Michael (2018), "Hurwitz's regular map (3,7) of genus 7: a polyhedral realization", The Art of Discrete and Applied Mathematics, 1 (1), Paper No. 1.02, doi:10.26493/2590-9770.1186.258, MR3995533.
Bujalance, Emilio; Costa, Antonio F. (1994), "Study of the symmetries of the Macbeath surface", Mathematical contributions, Madrid: Editorial Complutense, pp. 375–385, MR1303808.
Macbeath, A. Murray (1999). "Hurwitz Groups and Surfaces"(PDF). In Levy, Silvio (ed.). The Eightfold Way: The Beauty of Klein's Quartic Curve. MSRI Publications. Vol. 35. Cambridge: Cambridge University Press. pp. 103–113. ISBN0-521-66066-1. Retrieved 30 March 2025.
Serre, J.-P. (1994). "A Letter as an Appendix to the Square-Root Parameterization Paper Of Abhyankar". In Bajaj, Chandrajit L. (ed.). Algebraic Geometry and its Applications: Collections of Papers from Shreeram S. Abhyankar’s 60th Birthday Conference. New York: Springer. pp. 85–88. doi:10.1007/978-1-4612-2628-4_3.
Serre, J.-P. (2000). Oeuvres - Collected Papers IV. Springer Collected Works in Mathematics. Heidelberg: Springer Berlin. pp. 349–353.
Wohlfahrt, K. (1985), "Macbeath's curve and the modular group", Glasgow Math. J., 27: 239–247, doi:10.1017/S0017089500006212, MR0819842. Corrigendum, vol. 28, no. 2, 1986, p. 241, MR0848433.
