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

in ${\displaystyle \mathbb {C} ^{2}}$. 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.

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 ${\displaystyle {\tfrac {\pi }{2}},{\tfrac {\pi }{3}},{\tfrac {\pi }{8}}}$. 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 ${\displaystyle s_{2},s_{3},s_{8}}$ and relations ${\displaystyle s_{2}{}^{2}=s_{3}{}^{3}=s_{8}{}^{8}=1}$ as well as ${\displaystyle s_{2}s_{3}=s_{8}}$. 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.

...
Wikipedia