Bolza surface

In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by Oskar Bolza (1887)), is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely GL₂(3) of order 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 Riemann 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

The tiling of the Bolza surface by reflection domains is a quotient of the order-3 bisected octagonal tiling.

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. 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

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

• Hyperelliptic curve
• Klein quartic
• Macbeath surface

References

• Bolza, Oskar (1887), "On Binary Sextics with Linear Transformations into Themselves", American Journal of Mathematics, 10 (1): 47–70, doi:10.2307/2369402, JSTOR 2369402
• 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. doi:10.2140/pjm.2006.227.95.
• Maclachlan, C.; Reid, A. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Math. 219. New York: Springer. ISBN 0-387-98386-4.