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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Bolza surface, alternatively, complex algebraic Bolza curve (introduced by ), is a compact
Riemann surface In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed ver ...
of
genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of living and fossil organisms as well as viruses. In the hierarchy of biological classification, genus comes above species and below family. In binomial nom ...
2 with the highest possible order of the conformal
automorphism group In mathematics, the automorphism group of an object ''X'' is the group consisting of automorphisms of ''X'' under composition of morphisms. For example, if ''X'' is a finite-dimensional vector space, then the automorphism group of ''X'' is the g ...
in this genus, namely GL_2(3) of order 48 (the
general linear group In mathematics, the general linear group of degree ''n'' is the set of invertible matrices, together with the operation of ordinary matrix multiplication. This forms a group, because the product of two invertible matrices is again invertible, ...
of 2\times 2 matrices over the
finite field In mathematics, a finite field or Galois field (so-named in honor of Évariste Galois) is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtr ...
\mathbb_3). The full automorphism group (including reflections) is the
semi-direct product In mathematics, specifically in group theory, the concept of a semidirect product is a generalization of a direct product. There are two closely related concepts of semidirect product: * an ''inner'' semidirect product is a particular way in w ...
GL_(3)\rtimes\mathbb_ 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 In algebraic geometry, the smooth completion (or smooth compactification) of a smooth affine algebraic curve ''X'' is a complete smooth algebraic curve which contains ''X'' as an open subset. Smooth completions exist and are unique over a perfect ...
of the affine curve. Of all genus 2 hyperbolic surfaces, the Bolza surface maximizes the length of the
systole Systole ( ) is the part of the cardiac cycle during which some chambers of the heart contract after refilling with blood. The term originates, via New Latin, from Ancient Greek (''sustolē''), from (''sustéllein'' 'to contract'; from ...
. 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 In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at ea ...
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 In mathematics, spectral theory is an inclusive term for theories extending the eigenvector and eigenvalue theory of a single square matrix to a much broader theory of the structure of operators in a variety of mathematical spaces. It is a result ...
of the
Laplace–Beltrami operator In differential geometry, the Laplace–Beltrami operator is a generalization of the Laplace operator to functions defined on submanifolds in Euclidean space and, even more generally, on Riemannian and pseudo-Riemannian manifolds. It is named ...
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 In linear algebra, an eigenvector () or characteristic vector of a linear transformation is a nonzero vector that changes at most by a scalar factor when that linear transformation is applied to it. The corresponding eigenvalue, often denote ...
of the Laplacian among all compact, closed
Riemann surfaces In mathematics, particularly in complex analysis, a Riemann surface is a connected one-dimensional complex manifold. These surfaces were first studied by and are named after Bernhard Riemann. Riemann surfaces can be thought of as deformed vers ...
of genus 2 with constant negative
curvature In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the can ...
.


Triangle surface

The Bolza surface is a (2,3,8) triangle surface – see
Schwarz triangle In geometry, a Schwarz triangle, named after Hermann Schwarz, is a spherical triangle that can be used to tile a sphere ( spherical tiling), possibly overlapping, through reflections in its edges. They were classified in . These can be define ...
. More specifically, the
Fuchsian group In mathematics, a Fuchsian group is a discrete subgroup of PSL(2,R). The group PSL(2,R) can be regarded equivalently as a group of isometries of the hyperbolic plane, or conformal transformations of the unit disc, or conformal transformations o ...
defining the Bolza surface is a subgroup of the group generated by reflections in the sides of a hyperbolic triangle with angles \tfrac, \tfrac, \tfrac. The group of orientation preserving isometries is a subgroup of the
index Index (or its plural form indices) may refer to: Arts, entertainment, and media Fictional entities * Index (''A Certain Magical Index''), a character in the light novel series ''A Certain Magical Index'' * The Index, an item on a Halo megastru ...
-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 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 and corners at :p_k=2^e^, 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=\begin1+\sqrt & (2+\sqrt)\alpha e^\\(2+\sqrt)\alpha e^ & 1+\sqrt\end, where \alpha=\sqrt and k=0,\ldots, 3, along with their inverses. The generators satisfy the relation :g_0 g_1^ g_2 g_3^ g_0^ g_1 g_2^ g_3=1. These generators are connected to the
length spectrum Length is a measure of distance. In the International System of Quantities, length is a quantity with dimension distance. In most systems of measurement a base unit for length is chosen, from which all other units are derived. In the In ...
, 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(1+\sqrt)\approx 3.05714. The n^\text element \ell_n of the length spectrum for the Bolza surface is given by :\ell_n=2\operatorname(m+n\sqrt), where n runs through the
positive integers In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called '' cardinal ...
(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\vert. It is possible to obtain an equivalent closed form of the systole directly from the triangle group.
Formulae In science, a formula is a concise way of expressing information symbolically, as in a mathematical formula or a '' chemical formula''. The informal use of the term ''formula'' in science refers to the general construct of a relationship betw ...
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\left(\tfrac\right)\approx 3.05714. The geodesic lengths \ell_n also appear in the
Fenchel–Nielsen coordinates In mathematics, Fenchel–Nielsen coordinates are coordinates for Teichmüller space introduced by Werner Fenchel and Jakob Nielsen. Definition Suppose that ''S'' is a compact Riemann surface of genus Genus ( plural genera ) is a taxonomic ra ...
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;\; \ell_1,0;\; \ell_1,0), where \ell_2=2\operatorname(3+2\sqrt)\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\approx 0.321281.


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: : \langle R,\,S,\,T,\,U\mid 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, 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.


Spectral theory

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 In mathematics, the Teichmüller space T(S) of a (real) topological (or differential) surface S, is a space that parametrizes complex structures on S up to the action of homeomorphisms that are isotopic to the identity homeomorphism. Teichmüll ...
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 \zeta(-1/2) of the Bolza surface are :\det_(\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.


Quaternion algebra

Following MacLachlan and Reid, the
quaternion algebra In mathematics, a quaternion algebra over a field ''F'' is a central simple algebra ''A'' over ''F''See Milies & Sehgal, An introduction to group rings, exercise 17, chapter 2. that has dimension 4 over ''F''. Every quaternion algebra becomes a ma ...
can be taken to be the algebra over \mathbb(\sqrt) generated as an associative algebra by generators ''i,j'' and relations :i^2=-3,\;j^2=\sqrt,\;ij=-ji, with an appropriate choice of an
order Order, ORDER or Orders may refer to: * Categorization, the process in which ideas and objects are recognized, differentiated, and understood * Heterarchy, a system of organization wherein the elements have the potential to be ranked a number of ...
.


See also

*
Hyperelliptic curve In algebraic geometry, a hyperelliptic curve is an algebraic curve of genus ''g'' > 1, given by an equation of the form y^2 + h(x)y = f(x) where ''f''(''x'') is a polynomial of degree ''n'' = 2''g'' + 1 > 4 or ''n'' = 2''g'' + 2 > 4 with ''n'' dis ...
*
Klein quartic In hyperbolic geometry, the Klein quartic, named after Felix Klein, is a compact Riemann surface of genus with the highest possible order automorphism group for this genus, namely order orientation-preserving automorphisms, and automorphisms ...
*
Bring's curve In mathematics, Bring's curve (also called Bring's surface) is the curve given by the equations :v+w+x+y+z=v^2+w^2+x^2+y^2+z^2=v^3+w^3+x^3+y^3+z^3=0. It was named by after Erland Samuel Bring who studied a similar construction in 1786 in a Promot ...
* Macbeath surface *
First Hurwitz triplet In the mathematical theory of Riemann surfaces, the first Hurwitz triplet is a triple of distinct Hurwitz surfaces with the identical automorphism group of the lowest possible genus, namely 14 (genera 3 and 7 each admit a unique Hurwitz surface, re ...


References

* * * * * * * * ;Specific {{Systolic geometry navbox Riemann surfaces Systolic geometry