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In
mathematics, Bring's curve (also called Bring's surface) is the
curve
In mathematics, a curve (also called a curved line in older texts) is an object similar to a line, but that does not have to be straight.
Intuitively, a curve may be thought of as the trace left by a moving point. This is the definition that ...
given by the equations
:
It was named by after
Erland Samuel Bring
Erland Samuel Bring (19 August 1736 – 20 May 1798) was a Swedish mathematician.
Bring studied at Lund University between 1750 and 1757. In 1762 he obtained a position of a reader in history and was promoted to professor in 1779. At Lund he wr ...
who studied a similar construction in 1786 in a Promotionschrift submitted to the
University of Lund
, motto = Ad utrumque
, mottoeng = Prepared for both
, established =
, type = Public research university
, budget = SEK 9 billion [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 ...]
of the curve is the
symmetric group
In abstract algebra, the symmetric group defined over any set is the group whose elements are all the bijections from the set to itself, and whose group operation is the composition of functions. In particular, the finite symmetric group ...
''S''
5 of
order 120, given by
permutations of the 5 coordinates. This is the largest possible automorphism group of a genus 4 complex curve.
The curve can be realized as a
triple cover of the sphere branched in 12 points, and is the
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 ...
associated to the
small stellated dodecahedron
In geometry, the small stellated dodecahedron is a Kepler-Poinsot polyhedron, named by Arthur Cayley, and with Schläfli symbol . It is one of four nonconvex regular polyhedra. It is composed of 12 pentagrammic faces, with five pentagrams meeti ...
. It has genus 4. The full group of symmetries (including reflections) is the direct product
, which has order 240.
Fundamental domain and systole
Bring's curve can be obtained as a Riemann surface by associating sides of a hyperbolic
icosagon (see
fundamental polygon In mathematics, a fundamental polygon can be defined for every compact Riemann surface of genus greater than 0. It encodes not only the topology of the surface through its fundamental group but also determines the Riemann surface up to conformal eq ...
). The identification pattern is given in the adjoining diagram. The icosagon (of area
, by the
Gauss-Bonnet theorem) can be tessellated by 240 (2,4,5) triangles. The actions that transport one of these triangles to another give the full group of automorphisms of the surface (including reflections). Discounting reflections, we get the 120 automorphisms mentioned in the introduction. Note that 120 is less than 252, the maximum number of orientation preserving automorphisms allowed for a genus 4 surface, by
Hurwitz's automorphism theorem. Therefore, Bring's surface is not a
Hurwitz surface
In Riemann surface theory and hyperbolic geometry, a Hurwitz surface, named after Adolf Hurwitz, is a compact Riemann surface with precisely 84(''g'' − 1) automorphisms, where ''g'' is the genus of the surface. This number is maximal by vir ...
. This also tells us that there does not exist a Hurwitz surface of genus 4.
![Fundamental domain of Bring curve](https://upload.wikimedia.org/wikipedia/commons/b/b1/Fundamental_domain_of_Bring_curve.svg)
The full group of symmetries has the following presentation:
:
,
where
is the identity action,
is a rotation of order 5 about the centre of the fundamental polygon,
is a rotation of order 2 at the vertex where 4 (2,4,5) triangles meet in the tessellation, and
is reflection in the real line. From this presentation, information about the linear
representation theory
Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essen ...
of the symmetry group of Bring's surface can be computed using
GAP. In particular, the group has four 1 dimensional, four 4 dimensional, four 5 dimensional, and two 6 dimensional irreducible representations, and we have
:
as expected.
The
systole of the surface has length
:
Similarly to the
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 surface does not maximize the systole length among compact Riemann surfaces in its topological category (that is, surfaces having the same genus) despite maximizing the size of the automorphism group. The systole is presumably maximized by the surface referred to a M4 in . The systole length of M4 is
:
and has multiplicity 36.
Spectral theory
Little is known about 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 Bring's surface, however, it could potentially be of interest in this field. The
Bolza 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 ...
and Klein quartic have the largest symmetry groups among compact Riemann surfaces of constant negative curvature in genera 2 and 3 respectively, and thus it has been conjectured that they maximize the first positive eigenvalue in the Laplace spectrum. There is strong numerical evidence to support this hypothesis, particularly in the case of the Bolza surface, although providing a rigorous proof is still an open problem. Following this pattern, one may reasonably conjecture that Bring's surface maximizes the first positive eigenvalue of the Laplacian (among surfaces in its topological class).
See also
*
Bolza 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 ...
*
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 ...
*
Macbeath surface In Riemann surface theory and hyperbolic geometry, the Macbeath surface, also called Macbeath's curve or the Fricke–Macbeath curve, is the genus-7 Hurwitz surface.
The automorphism group of the Macbeath surface is the simple group PSL(2,8), con ...
*
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, res ...
References
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