In
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 János Bolyai, Bolyai–Nikolai Lobachevsky, Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with:
:For a ...
, the Macbeath surface, also called Macbeath's curve or the Fricke–Macbeath curve, is the genus-7
Hurwitz surface.
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 orientation-preserving isometries of the hyperbolic plane, or conformal transformations of the unit disc, or co ...
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 for its connection to Hurwitz surfaces, namely Riemann surfaces of genus ''g'' with the largest possible order, 84(''g'' − 1), of it ...
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
Systole ( ) is the part of the cardiac cycle during which some chambers of the heart contract after refilling with blood. Its contrasting phase is diastole, the relaxed phase of the cardiac cycle when the chambers of the heart are refilling ...
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 (: polyhedra or polyhedrons; ) is a three-dimensional figure with flat polygonal Face (geometry), faces, straight Edge (geometry), edges and sharp corners or Vertex (geometry), vertices. The term "polyhedron" may refer ...
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". In a later survey article Macbeath attributes the result to Fricke.
[, .]
See also
*
Klein quartic
*
First Hurwitz triplet
Notes
References
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*. Translation in ''Moscow Univ. Math. Bull.'' 44 (1989), no. 5, 37–40.
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*. Corrigendum, vol. 28, no. 2, 1986, p. 241, .
{{refend
Hyperbolic geometry
Riemann surfaces
Riemannian geometry
Differential geometry of surfaces
Systolic geometry