The surface'sFuchsian 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 andHurwitz 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. Itssystole is about 5.796, and the number of systolic loops is 126 according to R. Vogeler's calculations.
It is possible to realize the resulting triangulated surface as a non-convexpolyhedron without self-intersections.[2]
This surface was originally discovered byRobert Fricke (1899), but named afterAlexander Murray Macbeath due to his later independent rediscovery of the same curve.[3] Elkies writes that the equivalence between the curves studied by Fricke and Macbeath "may first have been observed bySerre in a 24.vii.1990 letter toAbhyankar".[4]
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,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.
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,MR0819842. Corrigendum, vol. 28, no. 2, 1986, p. 241,MR0848433.
