Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...

theory and

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

, the Macbeath surface, also called Macbeath's curve or the Fricke–Macbeath curve, is the genus-7

Hurwitz surface In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(''g'' − 1) automorphisms, where ''g'' is the genus of the surface. This number is maximal by virt ...

. The

automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...

of the Macbeath surface is the

simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...

PSL(2,8), consisting of 504 symmetries.

Triangle group construction

The surface's

Fuchsian group In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations of t ...

can be constructed as the principal congruence subgroup of the

(2,3,7) triangle group In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus ''g'' with the largest possible order, ...

in a suitable tower of principal congruence subgroups. Here the choices of quaternion algebra and

Hurwitz quaternion order The Hurwitz quaternion order is a specific order in a quaternion algebra over a suitable number field. The order is of particular importance in Riemann surface theory, in connection with surfaces with maximal symmetry, namely the Hurwitz surfaces. ...

are described at the triangle group page. Choosing the ideal

\langle 2 \rangle

in the ring of integers, the corresponding principal congruence subgroup defines this surface of genus 7. Its

systole Systole ( ) is the part of the cardiac cycle during which some chambers of the heart contract after refilling with blood. The term originates, via New Latin, from Ancient Greek (''sustolē''), from (''sustéllein'' 'to contract'; from ''sun ...

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 In geometry, a polyhedron (plural polyhedra or polyhedrons; ) is a three-dimensional shape with flat polygonal faces, straight edges and sharp corners or vertices. A convex polyhedron is the convex hull of finitely many points, not all on th ...

without self-intersections.

Historical note

This surface was originally discovered by , but named after

Alexander Murray Macbeath Alexander Murray Macbeath (30 June 1923 Glasgow – 14 May 2014 Warwick) was a mathematician who worked on Riemann surfaces. Macbeath surfaces and Macbeath regions are named after him. Early life and education Macbeath was the son of Alexande ...

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".

Notes

References

*. *. *. *. *. *. Translation in ''Moscow Univ. Math. Bull.'' 44 (1989), no. 5, 37–40. *. *. *. Corrigendum, vol. 28, no. 2, 1986, p. 241, . {{refend Hyperbolic geometry Riemann surfaces Riemannian geometry Differential geometry of surfaces Systolic geometry

Triangle group construction

Historical note

See also

Notes

References