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 vers ...

theory and

hyperbolic geometry In mathematics, hyperbolic geometry (also called Lobachevskian geometry or Bolyai– Lobachevskian geometry) is a non-Euclidean geometry. The parallel postulate of Euclidean geometry is replaced with: :For any given line ''R'' and point ''P'' ...

, a Hurwitz surface, named after

Adolf Hurwitz Adolf Hurwitz (; 26 March 1859 – 18 November 1919) was a German mathematician who worked on algebra, analysis, geometry and number theory. Early life He was born in Hildesheim, then part of the Kingdom of Hanover, to a Jewish family and died ...

, 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 versio ...

with precisely 84(''g'' − 1) automorphisms, where ''g'' is the

genus Genus ( plural genera ) is a taxonomic rank used in the biological classification of extant taxon, living and fossil organisms as well as Virus classification#ICTV classification, viruses. In the hierarchy of biological classification, genus com ...

of the surface. This number is maximal by virtue of Hurwitz's theorem on automorphisms . They are also referred to as Hurwitz curves, interpreting them as complex algebraic curves (complex dimension 1 = real dimension 2). 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 of t ...

of a Hurwitz surface is a

finite index In mathematics, specifically group theory, the index of a subgroup ''H'' in a group ''G'' is the number of left cosets of ''H'' in ''G'', or equivalently, the number of right cosets of ''H'' in ''G''. The index is denoted , G:H, or :H/math> or (G ...

torsionfree normal subgroup of the (ordinary)

(2,3,7) triangle group In the theory of Riemann surfaces and hyperbolic geometry, the triangle group (2,3,7) is particularly important. This importance stems from its connection to Hurwitz surfaces, namely Riemann surfaces of genus ''g'' with the largest possible order, ...

. The finite quotient group is precisely the automorphism group. Automorphisms of complex algebraic curves are ''

orientation-preserving The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, right-handed ...

'' automorphisms of the underlying real surface; if one allows orientation-''reversing'' isometries, this yields a group twice as large, of order 168(''g'' − 1), which is sometimes of interest. A note on terminology – in this and other contexts, the "(2,3,7) triangle group" most often refers, not to the ''full'' triangle group Δ(2,3,7) (the

Coxeter group In mathematics, a Coxeter group, named after H. S. M. Coxeter, is an abstract group that admits a formal description in terms of reflections (or kaleidoscopic mirrors). Indeed, the finite Coxeter groups are precisely the finite Euclidean refl ...

with

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 defined mor ...

(2,3,7) or a realization as a hyperbolic

reflection group In group theory and geometry, a reflection group is a discrete group which is generated by a set of reflections of a finite-dimensional Euclidean space. The symmetry group of a regular polytope or of a tiling of the Euclidean space by congruent cop ...

), but rather to the ''ordinary'' triangle group (the

von Dyck group In mathematics, a triangle group is a group that can be realized geometrically by sequences of reflections across the sides of a triangle. The triangle can be an ordinary Euclidean triangle, a triangle on the sphere, or a hyperbolic triangl ...

) ''D''(2,3,7) of orientation-preserving maps (the rotation group), which is index 2. The group of complex automorphisms is a quotient of the ''ordinary'' (orientation-preserving) triangle group, while the group of (possibly orientation-reversing) isometries is a quotient of the ''full'' triangle group.

Classification by genus

Only finitely many Hurwitz surfaces occur with each genus. The function

h(g)

mapping the genus to the number of Hurwitz surfaces with that genus is unbounded, even though most of its values are zero. The sum :

\sum_^\frac

converges for

s > 1/3

, implying in an approximate sense that the genus of the

n

th Hurwitz surface grows at least as a cubic function of

n

. The Hurwitz surface of least genus is 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 ...

of genus 3, with automorphism group the

projective special linear group In mathematics, especially in the group theoretic area of algebra, the projective linear group (also known as the projective general linear group or PGL) is the induced action of the general linear group of a vector space ''V'' on the associate ...

PSL(2,7) In mathematics, the projective special linear group , isomorphic to , is a finite simple group that has important applications in algebra, geometry, and number theory. It is the automorphism group of the Klein quartic as well as the symmetry group ...

, of order 84(3 − 1) = 168 = 2³·3·7, which is a

simple group SIMPLE Group Limited is a conglomeration of separately run companies that each has its core area in International Consulting. The core business areas are Legal Services, Fiduciary Activities, Banking Intermediation and Corporate Service. The d ...

; (or order 336 if one allows orientation-reversing isometries). The next possible genus is 7, possessed by the

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 Projective line ...

, with automorphism group PSL(2,8), which is the simple group of order 84(7 − 1) = 504 = 2³·3²·7; if one includes orientation-reversing isometries, the group is of order 1,008. An interesting phenomenon occurs in the next possible genus, namely 14. Here there is a triple of distinct Riemann surfaces with the identical automorphism group (of order 84(14 − 1) = 1092 = 2²·3·7·13). The explanation for this phenomenon is arithmetic. Namely, in the

ring of integers In mathematics, the ring of integers of an algebraic number field K is the ring of all algebraic integers contained in K. An algebraic integer is a root of a monic polynomial with integer coefficients: x^n+c_x^+\cdots+c_0. This ring is often deno ...

of the appropriate

number field In mathematics, an algebraic number field (or simply number field) is an extension field K of the field of rational numbers such that the field extension K / \mathbb has finite degree (and hence is an algebraic field extension). Thus K is a f ...

, the rational prime 13 splits as a product of three distinct

prime ideal In algebra, a prime ideal is a subset of a ring that shares many important properties of a prime number in the ring of integers. The prime ideals for the integers are the sets that contain all the multiples of a given prime number, together with ...

s. The

principal congruence subgroup In mathematics, a congruence subgroup of a matrix group with integer entries is a subgroup defined by congruence conditions on the entries. A very simple example would be invertible 2 × 2 integer matrices of determinant 1, in which the ...

s defined by the triplet of primes produce

s corresponding to the

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 ...

. The sequence of allowable values for the genus of a Hurwitz surface begins :3, 7, 14, 17, 118, 129, 146, 385, 411, 474, 687, 769, 1009, 1025, 1459, 1537, 2091, ...

References

*Elkies, N.: Shimura curve computations. ''Algorithmic number theory'' (Portland, OR, 1998), 1–47, Lecture Notes in Computer Science, 1423, Springer, Berlin, 1998. See * * Katz, M.; Schaps, M.; Vishne, U.: Logarithmic growth of

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 ''sun ...

of arithmetic Riemann surfaces along congruence subgroups. J. Differential Geom. 76 (2007), no. 3, 399-422. Available at * * {{Algebraic curves navbox Riemann surfaces Systolic geometry

Classification by genus

See also

References