Bolza surface

 ( Riemann Surfaces ) 

In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by ), is a compact Riemann surface of genus $2$ with the highest possible order of the conformal map automorphism group in this genus, namely $GL\_2(3)$ of order 48 (the general linear group of $2\backslash times\; 2$ matrices over the finite field $\backslash mathbb\{F\}\_3$). Its full automorphism group (including reflections) is the semi-direct product $GL\_\{2\}(3)\backslash rtimes\backslash mathbb\{Z\}\_\{2\}$ of order 96. An affine model for the Bolza surface can be obtained as the locus of the equation

$y^2=x^5-x$

in $\backslash mathbb\; C^2$. 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 $x\; =\; 0,\; 1,\; i,\; -1,\; -i,\; \backslash infty$.

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. 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 $2$ hyperbolic surfaces, the Bolza surface maximizes the length of the shortest closed geodesic, or systole .

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Triangle 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 $\backslash tfrac\{\backslash pi\}\{2\},\; \backslash tfrac\{\backslash pi\}\{3\},\; \backslash tfrac\{\backslash pi\}\{8\}$. 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 $\backslash Gamma$ 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 $\backslash Gamma$ on the Poincare disk, the fundamental domain of the Bolza surface is a regular octagon with angles $\backslash tfrac\{\backslash pi\}\{4\}$ and corners at

$p\_k=2^\{-1/4\}e^\{i\backslash left(\backslash tfrac\{\backslash pi\}\{8\}+\backslash tfrac\{k\backslash pi\}\{4\}\backslash right)\},$

where $k=0,\backslash ldots,\; 7$. Opposite sides of the octagon are identified under the action of the Fuchsian group. Its generators are the matrices

where $\backslash alpha=\backslash sqrt\{\backslash sqrt\{2\}-1\}$ and $k=0,\backslash ldots,\; 3$, 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

$\backslash ell\_1=2\backslash operatorname\{\backslash rm\; arcosh\}(1+\backslash sqrt\{2\})\backslash approx\; 3.05714.$

The $n^\backslash text\{th\}$ element $\backslash ell\_n$ of the length spectrum for the Bolza surface is given by

$\backslash ell\_n=2\backslash operatorname\{\backslash rm\; arcosh\}(m+n\backslash sqrt\{2\}),$

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

$\backslash vert\; m-n\backslash sqrt\{2\}\backslash vert.$

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,

$\backslash ell\_1=4\backslash operatorname\{\backslash rm\; arcosh\}\backslash left(\backslash tfrac\{\backslash csc\backslash left(\backslash tfrac\{\backslash pi\}\{8\}\backslash right)\}\{2\}\backslash right)\backslash approx\; 3.05714.$

The geodesic lengths $\backslash ell\_n$ 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 $(\backslash ell\_2,\backslash tfrac\{1\}\{2\};\backslash ;\; \backslash ell\_1,0;\backslash ;\; \backslash ell\_1,0)$, where $\backslash ell\_2=2\backslash operatorname\{\backslash rm\; arcosh\}(3+2\backslash sqrt\{2\})\backslash approx\; 4.8969$.

There is also a "symmetric" set of coordinates $(\backslash ell\_1,t;\backslash ;\; \backslash ell\_1,t;\backslash ;\; \backslash ell\_1,t)$, where all three of the lengths are the systole $\backslash ell\_1$ and all three of the twists are given by

$t=\backslash frac\{\backslash operatorname\{\backslash rm\; arcosh\}\backslash left(\backslash sqrt\{\backslash tfrac\{2\}\{7\}(3+\backslash sqrt\{2\})\}\backslash right)\}\{\backslash operatorname\{\backslash rm\; arcosh\}(1+\backslash sqrt\{2\})\}\backslash approx\; 0.321281.$

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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:

$\backslash langle\; R,\backslash ,S,\backslash ,T,\backslash ,U\backslash mid\; R^8=S^2=T^2=U^3=RSRS=STST=RTR^3\; T=e,\; \backslash ,UR=R^7\; U^2,\backslash ,U^2\; R=STU,\backslash ,US=SU^2,\backslash ,\; UT=RSU\; \backslash rangle,$

where $e$ is the trivial (identity) 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, and

$4(1^2)+2(2^2)+4(3^2)+3(4^2)=96$

as expected.

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Spectral theory Here, spectral theory refers to the spectrum of the Laplacian, $\backslash Delta$. 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.

+ Numerical computations of the first ten positive eigenvalues of the Bolza surface
$\backslash lambda\_0$	0	1
$\backslash lambda\_1$	3.8388872588421995185866224504354645970819150157	3
$\backslash lambda\_2$	5.353601341189050410918048311031446376357372198	4
$\backslash lambda\_3$	8.249554815200658121890106450682456568390578132	2
$\backslash lambda\_4$	14.72621678778883204128931844218483598373384446932	4
$\backslash lambda\_5$	15.04891613326704874618158434025881127570452711372	3
$\backslash lambda\_6$	18.65881962726019380629623466134099363131475471461	3
$\backslash lambda\_7$	20.5198597341420020011497712606420998241440266544635	4
$\backslash lambda\_8$	23.0785584813816351550752062995745529967807846993874	1
$\backslash lambda\_9$	28.079605737677729081562207945001124964945310994142	3
$\backslash lambda\_\{10\}$	30.833042737932549674243957560470189329562655076386	4

The spectral determinant and Casimir energy $\backslash zeta(-1/2)$ of the Bolza surface are

$\backslash det\{\}\_\{\backslash zeta\}(\backslash Delta)\backslash approx\; 4.72273280444557$

and

$\backslash zeta\_\backslash Delta(-1/2)\backslash approx\; -0.65000636917383$

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.

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Jacobian The Jacobian variety of the Bolza curve is the product of two copies of the elliptic curve $\backslash mathbb\{C\}/\backslash mathbb\{Z\}\backslash sqrt\{-2\}$.

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Quaternion algebra The quaternion algebra describing the $(3,3,4)$ triangle group can be taken to be the algebra over $\backslash mathbb\{Q\}(\backslash sqrt\{2\})$ generated as an associative algebra by generators i,j and relations

$i^2=-3,\backslash ;j^2=\backslash sqrt\{2\},\backslash ;ij=-ji,$

with an appropriate choice of an order.

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See also

• Hyperelliptic curve
• Klein quartic
• Bring's curve
• Macbeath surface
• First Hurwitz triplet

• (2025). 9780387983868, Springer. ISBN 9780387983868

Specific

Categories: Riemann Surfaces, Systolic Geometry

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