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.
It is possible to realize the resulting triangulated surface as a non-convex
polyhedron without self-intersections.[2]
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.[3] 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".[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.