Bolza surface

In mathematics, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by Oskar Bolza (1887)), is a compact Riemann surface of genus 2 with the highest possible order of the conformal automorphism group in this genus, namely GL₂(3) of order 48. The full automorphism group (including reflections) is the semi-direct product $GL_{2}(3)\rtimes \mathbb {Z} _{2}$ 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 $\mathbb {C} ^{2}$. The Bolza surface is the smooth completion of the affine curve. Of all genus 2 hyperbolic surfaces, the Bolza surface has the highest 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 tiling of the Bolza surface by reflection domains is a quotient of the order-3 bisected octagonal tiling.

The Bolza surface is 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 ${\tfrac {\pi }{2}},{\tfrac {\pi }{3}},{\tfrac {\pi }{8}}$. More specifically, it 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 realisation in terms 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

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

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

$g_{0}g_{1}^{-1}g_{2}g_{3}^{-1}g_{0}^{-1}g_{1}g_{2}^{-1}g_{3}=1.$

The fundamental domain of the Bolza surface in the Poincare disk; opposite sides are identified.

These generators are connected to the length spectrum. We have

$\cosh \left({\tfrac {l_{0}}{2}}\right)=1+{\sqrt {2}},$

where $l_{0}$ is the length of the systole. All other geodesics in the spectrum take the form

$\cosh \left({\tfrac {l_{n-1}}{2}}\right)=m+{\sqrt {2}}n,$

where $m,\,n\in \mathbb {N}$, $n$ runs through the natural numbers (apart from 48 and 72), and $m$ is the unique odd number that minimises

$\vert m/n-{\sqrt {2}}\vert .$

Here we have already seen a closed formula for the systole length. It is possible to obtain an equivalent form 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,

$4\cosh ^{-1}\left({\tfrac {\csc \left({\tfrac {\pi }{8}}\right)}{2}}\right)\approx 3.05714183896.$

The systole also appears in the Fenchel-Nielsen coordinates of the surface; in particular, one set of coordinates is

$\left(l_{1},t_{1};l_{2},t_{2};l_{3},t_{3}\right)=\left(2\cosh ^{-1}(3+2{\sqrt {2}}),{\tfrac {1}{2}};2\cosh ^{-1}(1+{\sqrt {2}}),0;2\cosh ^{-1}(1+{\sqrt {2}}),0\right).$

There is also a "symmetric" set of coordinates where all the length parameters are given by the systole, and have equal twist parameters, namely

$\left(l_{1},t_{1};l_{2},t_{2};l_{3},t_{3}\right)=\left(2\cosh ^{-1}(1+{\sqrt {2}}),{\tfrac {\sqrt {{\sqrt {2}}-1}}{2}};2\cosh ^{-1}(1+{\sqrt {2}}),{\tfrac {\sqrt {{\sqrt {2}}-1}}{2}};2\cosh ^{-1}(1+{\sqrt {2}}),{\tfrac {\sqrt {{\sqrt {2}}-1}}{2}}\right).$

Symmetries of the surface

The four generators of the symmetry group of the Bolza surface

When we view the fundamental domain of the Bolza surface as 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:

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

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, we find that there are four 1 dimensional, two 2 dimensional, four 3 dimensional, and three 4 dimensional irreducible representations. Indeed, we can check that these satisfy Burnside's lemma, that is

$4(1^{2})+2(2^{2})+4(3^{2})+3(4^{2})=96,$

which is the order of the group (including reflections) as expected. These representations are investigated further in (Cook 2018), where group theoretic and analytical methods are combined to prove results about the Laplace spectrum.

Spectral theory

Finite element plots of the three eigenfunctions corresponding to the first positive eigenvalue of the Bolza surface. Functions are zero on the light blue lines.

Here we disucss the spectral theory of the Laplacian, $\Delta$. The first eigenspace (that is, the eigenspace corresponding to the first positive eigenvalue) of the Bolza surface has been shown to be three dimensional, and the second, four dimensional (Cook 2018), (Jenni 1981). 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 has been computed to a very high accuracy by Strohmaier & Uski (2013), where the authors present an algorithm for computing the eigenvalues of the Laplace operator on hyperbolic surfaces with a given precision. The algorithm is based on an adaption of the method of particular solutions to the case of locally symmetric spaces. The following table gives the first ten positive eigenvalues of the Bolza surface, computed using this method; more eigenvalues have been computed and these can be found as an ancillary file on the corresponding arXiv page.

Numerical computations of the first ten positive eigenvalues of the Bolza surface

Eigenvalue	Numerical value	Multiplicity
$\lambda _{0}$	0	1
$\lambda _{1}$	3.8388872588421995185866224504354645970819150157	3
$\lambda _{2}$	5.353601341189050410918048311031446376357372198	4
$\lambda _{3}$	8.249554815200658121890106450682456568390578132	2
$\lambda _{4}$	14.72621678778883204128931844218483598373384446932	4
$\lambda _{5}$	15.04891613326704874618158434025881127570452711372	3
$\lambda _{6}$	18.65881962726019380629623466134099363131475471461	3
$\lambda _{7}$	20.5198597341420020011497712606420998241440266544635	4
$\lambda _{8}$	23.0785584813816351550752062995745529967807846993874	1
$\lambda _{9}$	28.079605737677729081562207945001124964945310994142	3
$\lambda _{10}$	30.833042737932549674243957560470189329562655076386	4

Strohmaier and Uski also computed the spectral determinant and Casimir energy $\zeta (-1/2)$ of the surface. They 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. The authors conjecture 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 $\mathbb {Q} ({\sqrt {2}})$ 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

• Hyperelliptic curve
• Klein quartic
• Macbeath surface

References

• Bolza, Oskar (1887), "On Binary Sextics with Linear Transformations into Themselves", American Journal of Mathematics, 10 (1): 47–70, doi:10.2307/2369402, JSTOR 2369402 {{citation}}: Cite has empty unknown parameter: |1= (help)
• Katz, M.; Sabourau, S. (2006). "An optimal systolic inequality for CAT(0) metrics in genus two". Pacific J. Math. 227 (1): 95–107. arXiv:math.DG/0501017. doi:10.2140/pjm.2006.227.95.
• Cook, J. (2018). Properties of Eigenvalues on Riemann Surfaces with Large Symmetry Groups (PhD thesis, unpublished). Loughborough University. {{cite thesis}}: Invalid |ref=harv (help)
• Jenni, F. (1981). Ueber das Spektrum des Laplace-Operators auf einer Schar kompakter Riemannscher Flachen (PhD thesis). University of Basel. {{cite thesis}}: Invalid |ref=harv (help)
• Strohmaier, A.; Uski, V. (2013). "An Algorithm for the Computation of Eigenvalues, Spectral Zeta Functions and Zeta-Determinants on Hyperbolic Surfaces". Communications in Mathematical Physics. 317 (3): 827–869. arXiv:1110.2150v4. doi:10.1007/s00220-012-1557-1. {{cite journal}}: Invalid |ref=harv (help)
• Maclachlan, C.; Reid, A. (2003). The Arithmetic of Hyperbolic 3-Manifolds. Graduate Texts in Math. Vol. 219. New York: Springer. ISBN 0-387-98386-4.