Wikipedia / Bolza surface
Wikipedia / Bolza surface
In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by ), is a compact Riemann surface of genus with the highest possible order of the conformal automorphism group of order 48 (the general linear group matrices over the finite field). Its full automorphism group (including reflections) is a semi-direct product of order 96. (GAP identifies these two groups as SmallGroup(48,29) and SmallGroup(96,193).) An affine model for the Bolza surface can be obtained as the locus of the equation
. 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.
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 with constant negative curvature hyperbolic surfaces, the Bolza surface maximizes the length of the shortest closed geodesic, or systole .
Triangle surface
triangle surface – see Schwarz triangle. More specifically, the Fuchsian group. The group of orientation preserving isometries is a subgroup of the index defining the Bolza surface is also a subgroup of the (3,3,4) triangle group group does.
on the Poincaré disk and corners at
. Opposite sides of the octagon are identified under the action of the Fuchsian group. Its generators are the matrices
, along with their inverses. The generators satisfy the relation
These generators are connected to the length spectrum, which is the set of all the possible lengths of geodesic loops. The shortest such length is called the systole of the surface. The systole of the Bolza surface is
of the length spectrum for the Bolza surface is given by
runs through the positive integers (but omitting 4, 24, 48, 72, 140, and various higher values) 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. The systole is equal to four times the length of the side of medial length in a (2,3,8) triangle, that is,
also appear in the Fenchel–Nielsen coordinates.
and all three of the twists are given by
Symmetries of the surface
The symmetry of the Bolza surface can be understood in terms of its tiling by 16 equilateral hyperbolic triangles, 8 per vertex. Focusing on one particular choice among the 16 triangles, call it T, any symmetry of the Bolza surface must carry T to one of the 16 triangles (possibly itself). Call this triangle T'. Triangle T can map to T' in any of exactly 6 ways that preserve distance, one for each permutation of its vertices. This shows there are 6×16 = 96 symmetries in all.
One common choice of fundamental domain for 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:
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
as expected.
Another presentation of the full symmetry group starts with the triangle group. Letting s, l, and h denote flipping across the short leg, long leg, or hypotenuse of the triangle, the full symmetry group is
.
Spectral theory
. 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 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.
Jacobian
The Jacobian variety of the Bolza curve is the product of two copies of the elliptic curve.
Quaternion algebra
The quaternion algebra generated as an associative algebra by generators i,j and relations
with an appropriate choice of an order.
See also
Footnotes
References