Bolza surface


In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve, is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely GL2 of order 48. The full automorphism group is the semi-direct product of order 96. An affine model for the Bolza surface can be obtained as the locus of the equation
in. The Bolza surface is the smooth completion of the affine curve. Of all genus 2 hyperbolic surfaces, the Bolza surface maximizes the length of the 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 has attracted the attention of physicists, as it provides a relatively simple model for quantum chaos; in this context, it is usually referred to as the Hadamard–Gutzwiller model. The spectral theory of the Laplace–Beltrami operator acting on functions on the Bolza surface is of interest to both mathematicians and physicists, since the surface is conjectured to maximize the first positive eigenvalue of the Laplacian among all compact, closed Riemann surfaces of genus 2 with constant negative curvature.

Triangle surface

The Bolza surface is a 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. 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 and relations as well as. The Fuchsian group defining the Bolza surface is also a subgroup of the triangle group, which is a subgroup of index 2 in the triangle group. The group does not have a realization in terms of a quaternion algebra, but the group does.
Under the action of on the Poincare disk, the fundamental domain of the Bolza surface is a regular octagon with angles and corners at
where. Opposite sides of the octagon are identified under the action of the Fuchsian group. Its generators are the matrices
where and, along with their inverses. The generators satisfy the relation
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
The element of the length spectrum for the Bolza surface is given by
where runs through the positive integers and where is the unique odd integer that minimizes
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 triangles explicitly. The systole is equal to four times the length of the side of medial length in a triangle, that is,
The geodesic lengths 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, where.
There is also a "symmetric" set of coordinates, where all three of the lengths are the systole and all three of the twists
are given by

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 symmetry group are:
These are shown by the bold lines in the adjacent figure. They satisfy the following set of relations:
where is the trivial action. One may use this set of relations in GAP to retrieve information about the representation theory of the group. In particular, there are four 1-dimensional, two 2-dimensional, four 3-dimensional, and three 4-dimensional irreducible representations. Indeed, these values satisfy Burnside's lemma, that is
which is the order of the group as expected.

Spectral theory

Here, spectral theory refers to the spectrum of the Laplacian,. The first eigenspace 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.
EigenvalueNumerical valueMultiplicity
01
3.83888725884219951858662245043546459708191501573
5.3536013411890504109180483110314463763573721984
8.2495548152006581218901064506824565683905781322
14.726216787788832041289318442184835983733844469324
15.048916133267048746181584340258811275704527113723
18.658819627260193806296234661340993631314754714613
20.51985973414200200114977126064209982414402665446354
23.07855848138163515507520629957455299678078469938741
28.0796057376777290815622079450011249649453109941423
30.8330427379325496742439575604701893295626550763864

The spectral determinant and Casimir energy of the Bolza surface are
and
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.

Quaternion algebra

Following MacLachlan and Reid, the quaternion algebra can be taken to be the algebra over generated as an associative algebra by generators i,j and relations
with an appropriate choice of an order.