MacBeath surface explained

In Riemann surface theory and hyperbolic geometry, the MacBeath surface, also called MacBeath curve or the Fricke–MacBeath surface curve, is the genus-7 Hurwitz surface. It is named for Murray MacBeath.

The automorphism group of the MacBeath surface is the simple group PSL(2,8), consisting of 504 symmetries.

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

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.

Historical note

This surface was originally discovered by, but named after Alexander Murray Macbeath due to his later independent rediscovery of the same curve. 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".^[1] In a later survey article Macbeath attributes the result to Fricke.^[2]

See also

• Klein quartic
• First Hurwitz triplet

References

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• . Translation in Moscow Univ. Math. Bull. 44 (1989), no. 5, 37–40.
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• Macbeath . A. Murray . Hurwitz Groups and Surfaces . The Eightfold Way: The Beauty of Klein's Quartic Curve . Levy . Silvio . MSRI Publications . 35 . 1999 . Cambridge University Press . Cambridge . 0-521-66066-1 . 103–113 . 30 March 2025.
• Vogeler. Roger. On the geometry of Hurwitz surfaces. PhD. Florida State University. 2003. 29 March 2025. .
• Serre. J.-P.. A Letter as an Appendix to the Square-Root Parameterization Paper Of Abhyankar . Bajaj . Chandrajit L. . Algebraic Geometry and its Applications: Collections of Papers from Shreeram S. Abhyankar’s 60th Birthday Conference . Springer . New York . 10.1007/978-1-4612-2628-4_3 . 1994 . 85–88.
• Book: Serre, J.-P. . Oeuvres - Collected Papers IV. Springer Berlin . Heidelberg . 2000 . Springer Collected Works in Mathematics . 349–353.
• . Corrigendum, vol. 28, no. 2, 1986, p. 241, .

Notes and References

1. . The letter is reprinted in and .
2. , .