Bolza surface
with the highest possible order of the conformal automorphism group in this genus, namely
Its full automorphism group (including reflections) is the semi-direct product
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 this affine curve.
The Bolza curve also arises as a branched double cover of the Riemann sphere with branch points at the six vertices of a regular octahedron inscribed in the sphere.
This can be seen from the equation above, because the right-hand side becomes zero or infinite at the six points
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.
[1] 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
Eigenvectors of the Laplace-Beltrami operator are quantum analogues of periodic orbits, and as a classical analogue of this conjecture, it is known that of all genus
hyperbolic surfaces, the Bolza surface maximizes the length of the shortest closed geodesic, or systole (Schmutz 1993).
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
group does not have a realization in terms of a quaternion algebra, but the
on the Poincare disk, the fundamental domain of the Bolza surface is a regular octagon with angles
Opposite sides of the octagon are identified under the action of the Fuchsian group.
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.
runs through the positive integers (but omitting 4, 24, 48, 72, 140, and various higher values) (Aurich, Bogomolny & Steiner 1991) 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 (2,3,8) triangles explicitly.
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
and all three of the twists are given by[2] 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: These are shown by the bold lines in the adjacent figure.
One may use this set of relations in GAP to retrieve information about the representation theory of the group.
Here, spectral theory refers to the spectrum of the Laplacian,
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.
In particular, the spectrum of the Bolza surface is known to a very high accuracy (Strohmaier & Uski 2013).
The following table gives the first ten positive eigenvalues of the Bolza surface.
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.
