Bring's curve

Bring's curve can be obtained as a Riemann surface by associating sides of a hyperbolic icosagon (see fundamental polygon). The identification pattern is given in the adjoining diagram. The icosagon (of area ${\displaystyle 12\pi }$, by the Gauss-Bonnet theorem) can be tessellated by 240 (2,4,5) triangles. The actions that transport one of these triangles to another give the full group of automorphisms of the surface (including reflections). Discounting reflections, we get the 120 automorphisms mentioned in the introduction. Note that 120 is less that 252, the maximum number of orientation preserving automorphisms allowed for a genus 4 surface, by Hurwitz's automorphism theorem. Therefore, Bring's surface is not a Hurwitz surface. This also tells us that there does not exist a Hurwitz surface of genus 4.

The fundamental icosagon for Bring's surface, complete with side identifications.

The full group of symmetries has the following presentation:

${\displaystyle \langle r,\,s,\,t\,|\,r^{5}=s^{2}=t^{2}=rtrt=stst=(rs)^{4}=(sr^{3}sr^{2})^{2}=e\rangle }$,

where ${\displaystyle e}$ is the identity action, ${\displaystyle r}$ is a rotation of order 5 about the centre of the fundamental polygon, ${\displaystyle s}$ is a rotation of order 2 at the vertex where 4 (2,4,5) triangles meet in the tessellation, and ${\displaystyle t}$ is reflection in the real line. From this presentation, information about the linear representation theory of the symmetry group of Bring's surface can be computed using GAP. In particular, the group has four 1 dimensional, four 4 dimensional, four 5 dimensional, and two 6 dimensional irreducible representations, and we have

${\displaystyle 4(1^{2})+4(4^{2})+4(5^{2})+2(6^{2})=4+64+100+72=240}$

as expected.

The systole of the surface has length

${\displaystyle 12\sinh ^{-1}\left({\tfrac {1}{2}}{\sqrt {{\tfrac {1}{2}}({\sqrt {5}}-1)}}\right)\approx 4.60318.}$

Similarly to the Klein quartic, Bring's surface does not maximize the systole length among compact Riemann surfaces in its topological category (that is, surfaces having the same genus) despite maximizing the size of the automorphism group. The systole is presumably maximized by the surface referred to a M4 in (Schmutz 1993). The systole length of M4 is

${\displaystyle 2\cosh ^{-1}\left({\tfrac {1}{2}}(5+3{\sqrt {3}})\right)\approx 4.6245,}$

and has multiplicity 36.

Little is known about the spectral theory of Bring's surface, however, it could potentially be of interest in this field. The Bolza surface and Klein quartic have the largest symmetry groups among compact Riemann surfaces of constant negative curvature in genera 2 and 3 respectively, and thus it has been conjectured that they maximize the first positive eigenvalue in the Laplace spectrum. There is strong numerical evidence to support this hypothesis, particularly in the case of the Bolza surface, although providing a rigorous proof is still an open problem. Following this pattern, one may reasonably conjecture that Bring's surface maximizes the first positive eigenvalue of the Laplacian (among surfaces in its topological class).

• Bolza surface
• Klein quartic
• Macbeath surface
• First Hurwitz triplet

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