Bolza surface
In mathematics, a Riemann surface
title: "Bolza surface" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["riemann-surfaces", "systolic-geometry"] description: "In mathematics, a Riemann surface" topic_path: "science/mathematics" source: "https://en.wikipedia.org/wiki/Bolza_surface" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0
::summary In mathematics, a Riemann surface ::
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 automorphism group in this genus, namely GL_2(3) of order 48 (the general linear group of 2\times 2 matrices over the finite field \mathbb{F}3). Its full automorphism group (including reflections) is a semi-direct product GL{2}(3)\rtimes\mathbb{Z}_{2} 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
:y^2=x^5-x
in \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, \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 .
Triangle surface
::figure[src="https://upload.wikimedia.org/wikipedia/commons/7/7b/H2-8-3-kisrhombille.svg" caption="The tiling of the Bolza surface by reflection domains is a quotient of the [[order-3 bisected octagonal tiling]]."] ::
::figure[src="https://upload.wikimedia.org/wikipedia/commons/e/e0/Fundamental_domain_of_Bolza_surface.svg" caption="The fundamental domain of the Bolza surface in the Poincaré disk; opposite sides are identified."] ::
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 \tfrac{\pi}{2}, \tfrac{\pi}{3}, \tfrac{\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 \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 \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^{-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.
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
:\ell_1=2\operatorname{\rm arcosh}(1+\sqrt{2})\approx 3.05714.
The n^\text{th} element \ell_n of the length spectrum for the Bolza surface is given by
:\ell_n=2\operatorname{\rm arcosh}(m+n\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
:\vert m-n\sqrt{2}\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,
:\ell_1=4\operatorname{\rm arcosh}\left(\tfrac{\csc\left(\tfrac{\pi}{8}\right)}{2}\right)\approx 3.05714.
The geodesic lengths \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 (\ell_2,\tfrac{1}{2};; \ell_1,0;; \ell_1,0), where \ell_2=2\operatorname{\rm arcosh}(3+2\sqrt{2})\approx 4.8969.
There is also a "symmetric" set of coordinates (\ell_1,t;; \ell_1,t;; \ell_1,t), where all three of the lengths are the systole \ell_1 and all three of the twists are given by :t=\frac{\operatorname{\rm arcosh}\left(\sqrt{\tfrac{2}{7}(3+\sqrt{2})}\right)}{\operatorname{\rm arcosh}(1+\sqrt{2})}\approx 0.321281.
Symmetries of the surface
::figure[src="https://upload.wikimedia.org/wikipedia/commons/c/ca/Symmetries_of_the_Bolza_surface.png" caption="The four generators of the symmetry group of the Bolza 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:
: \langle R,,S,,T,,U\mid R^8=S^2=T^2=U^3=RSRS=STST=RURU=SUSU=R^7 USTU=R^4 UR^4 U^2 = e \rangle,
where e 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
:4(1^2)+2(2^2)+4(3^2)+3(4^2)=96
as expected.
Another presentation of the full symmetry group starts with the triangle group \Delta(2,3,8). Letting s, l, and h denote flipping across the short leg, long leg, or hypotenuse of the triangle, the full symmetry group is
: \langle s,l,h\mid s^2=l^2=h^2=(sl)^2=(sh)^3=(l h)^8=(sh(l h)^3)^2 = e \rangle,
where that last relation, the one that doesn't hold in the full triangle group, walks along a geodesic whose length is the systole \ell_1.
Spectral theory
::figure[src="https://upload.wikimedia.org/wikipedia/commons/9/9f/First_eigenspace_of_the_Bolza_surface.png" caption="Plots of the three eigenfunctions corresponding to the first positive eigenvalue of the Bolza surface. Functions are zero on the light blue lines. These plots were produced using [[FreeFEM++]]."] ::
Here, spectral theory refers to the spectrum of the Laplacian, \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.
::data[format=table title="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 |
| :: |
The spectral determinant and Casimir energy \zeta(-1/2) of the Bolza surface 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. 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 \mathbb{C}/\mathbb{Z}[\sqrt{-2}].
Quaternion algebra
The quaternion algebra describing the (3,3,4) triangle group 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.
References
;Specific
References
- (1 August 1988). "Quantum Chaos of the Hadamard–Gutzwiller Model". Physical Review Letters.
- (2017). "Computation of eigenvalues, spectral zeta functions and zeta-determinants on hyperbolic surfaces". Centre de Recherches Mathématiques and American Mathematical Society.
- (2021). "The Bolza curve and some orbifold ball quotient surfaces". Michigan Mathematical Journal.
- Katz, Karin. (2016-10-01). "Bolza Quaternion Order and Asymptotics of Systoles Along Congruence Subgroups". Experimental Mathematics.
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