Bolza surface

. The Bolza surface is conformally equivalent to 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}. The group of orientation preserving isometries 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 \Gamma 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 realization as the order-2 quotient of the group of norm-1 elements of a quaternion algebra, but the (3,3,4) group does. Under the action of \Gamma on the Poincare disk, the fundamental domain of the Bolza surface is a regular octagon with angles \tfrac{\pi}{4} and corners at :p_k=2^{-1/4}e^{i\left(\tfrac{\pi}{8}+\tfrac{k\pi}{4}\right)}, where k=0,\ldots, 7. Opposite sides of the octagon are identified under the action of the Fuchsian group. Its generators are the matrices :g_k=\begin{pmatrix}1+\sqrt{2} & (2+\sqrt{2})\alpha e^{\tfrac{ik\pi}{4}}\\(2+\sqrt{2})\alpha e^{ -\tfrac{ik\pi}{4}} & 1+\sqrt{2}\end{pmatrix}, where \alpha=\sqrt{\sqrt{2}-1} and k=0,\ldots, 3, along with their inverses. The generators satisfy the relation :g_0 g_1^{-1} g_2 g_3^{-1} g_0^{-1} g_1 g_2^{-1} g_3=1. These generators are connected to the length spectrum, which gives all of the possible lengths of geodesic loops. The shortest such length is called the systole of the surface. The systole of the Bolza surface is :\ell_1=2\operatorname{\rm arcosh}(1+\sqrt{2})\approx 3.05714. The n^\text{th} element \ell_n of the length spectrum for the Bolza surface is given by :\ell_n=2\operatorname{\rm arcosh}(m+n\sqrt{2}), where n runs through the positive integers (but omitting 4, 24, 48, 72, 140, and various higher values) and where m is the unique odd integer that minimizes :\vert m-n\sqrt{2}\vert. It is possible to obtain an equivalent closed form of the systole directly from the triangle group. Formulae exist to calculate the side lengths of a (2,3,8) triangles explicitly. The systole is equal to four times the length of the side of medial length in a (2,3,8) triangle, that is, :\ell_1=4\operatorname{\rm arcosh}\left(\tfrac{\csc\left(\tfrac{\pi}{8}\right)}{2}\right)\approx 3.05714. The geodesic lengths \ell_n also appear in the Fenchel–Nielsen coordinates of the surface. A set of Fenchel-Nielsen coordinates for a surface of genus 2 consists of three pairs, each pair being a length and twist. Perhaps the simplest such set of coordinates for the Bolza surface is (\ell_2,\tfrac{1}{2};\; \ell_1,0;\; \ell_1,0), where \ell_2=2\operatorname{\rm arcosh}(3+2\sqrt{2})\approx 4.8969. There is also a "symmetric" set of coordinates (\ell_1,t;\; \ell_1,t;\; \ell_1,t), where all three of the lengths are the systole \ell_1 and all three of the twists are given by :t=\frac{\operatorname{\rm arcosh}\left(\sqrt{\tfrac{2}{7}(3+\sqrt{2})}\right)}{\operatorname{\rm arcosh}(1+\sqrt{2})}\approx 0.321281. ==Symmetries of the surface==

The fundamental domain of the Bolza surface is a regular octagon in the Poincaré disk; the four symmetric actions that generate the (full) symmetry group are: • R – rotation of order 8 about the centre of the octagon; • S – reflection in the real line; • T – reflection in the side of one of the 16 (4,4,4) triangles that tessellate the octagon; • U – rotation of order 3 about the centre of a (4,4,4) triangle. These are shown by the bold lines in the adjacent figure. They satisfy the following set of relations: : \langle R,\,S,\,T,\,U\mid R^8=S^2=T^2=U^3=RSRS=STST=RURU=SUSU=R^7 USTU=R^4 UR^4 U^2 = e \rangle, where e is the trivial (identity) action. The first nine of those relations take the octagon shown to itself, but the last takes it to a neighboring octagon that shares an edge. One may use this set of relations in GAP to retrieve information about the representation theory of the group (which GAP identifies as SmallGroup(96,193)). In particular, there are four 1-dimensional, two 2-dimensional, four 3-dimensional, and three 4-dimensional irreducible representations, and :4(1^2)+2(2^2)+4(3^2)+3(4^2)=96 as expected. Another presentation of the full symmetry group starts with the triangle group \Delta(2,3,8). Letting s, l, and h denote flipping across the short leg, long leg, or hypotenuse of the triangle, the full symmetry group is : \langle s,l,h\mid s^2=l^2=h^2=(sl)^2=(sh)^3=(l h)^8=(sh(l h)^3)^2 = e \rangle, where that last relation, the one that doesn't hold in the full triangle group, walks along a geodesic whose length is the systole \ell_1. ==Spectral theory==

. Here, spectral theory refers to the spectrum of the Laplacian, \Delta. The first eigenspace (that is, the eigenspace corresponding to the first positive eigenvalue) of the Bolza surface is three-dimensional, and the second, four-dimensional , . It is thought that investigating perturbations of the nodal lines of functions in the first eigenspace in Teichmüller space will yield the conjectured result in the introduction. This conjecture is based on extensive numerical computations of eigenvalues of the surface and other surfaces of genus 2. In particular, the spectrum of the Bolza surface is known to a very high accuracy . The following table gives the first ten positive eigenvalues of the Bolza surface. The spectral determinant and Casimir energy \zeta(-1/2) of the Bolza surface are :\det{}_{\zeta}(\Delta)\approx 4.72273280444557 and :\zeta_\Delta(-1/2)\approx -0.65000636917383 respectively, where all decimal places are believed to be correct. It is conjectured that the spectral determinant is maximized in genus 2 for the Bolza surface. ==Jacobian==

The Jacobian variety of the Bolza curve is the product of two copies of the elliptic curve \mathbb{C}/\mathbb{Z}[\sqrt{-2}]. ==Quaternion algebra==

The quaternion algebra describing the (3,3,4) triangle group 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==