Bolza surface

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Perspective projection of y^2=x^5-x in \mathbb C^2.

In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (named after Oskar Bolza), is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely 48. An affine model for the Bolza surface can be obtained as the locus of the equation

y^2=x^5-x

in \mathbb C^2. The Bolza surface is the smooth completion of the affine curve. Of all genus 2 hyperbolic surfaces, the Bolza surface has the highest systole. As a hyperelliptic surface, it arises as the ramified double cover of the Riemann sphere, with ramification locus at the six vertices of a regular octahedron inscribed in the sphere, as can be readily seen from the equation above.

Triangle surface[edit]

The Bolza surface is a (2,3,8) triangle surface – see Schwarz triangle. More specifically, the Fuchsian group defining the Bolza surface is a subgroup of the group generated by reflections in the sides of a hyperbolic triangle with angles \tfrac{\pi}{2}, \tfrac{\pi}{3}, \tfrac{\pi}{8}. More specifically, it is a subgroup of the index-two subgroup of the group of reflections, which consists of products of an even number of reflections, which has an abstract presentation in terms of generators s_2, s_3, s_8 and relations s_2{}^2=s_3{}^3=s_8{}^8=1 as well as s_2 s_3 = s_8. The Fuchsian group defining the Bolza surface is also a subgroup of the (3,3,4) triangle group, which is a subgroup of index 2 in the (2,3,8) triangle group. It is interesting to note that the (2,3,8) group does not have a realisation in terms of a quaternion algebra, but the (3,3,4) group does.

Quaternion algebra[edit]

Following MacLachlan and Reid, the quaternion algebra can be taken to be the algebra over \mathbb{Q}(\sqrt{2}) generated as an associative algebra by generators i,j and relations

i^2=-3,\;j^2=\sqrt{2},\;ij=-ji,

with an appropriate choice of an order.

See also[edit]

References[edit]

  • Katz, M.; Sabourau, S. (2006). "An optimal systolic inequality for CAT(0) metrics in genus two". Pacific J. Math. 227 (1): 95–107. arXiv:math.DG/0501017. 
  • Maclachlan, C.; Reid, A. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Math. 219. New York: Springer. ISBN 0-387-98386-4.