Hurwitz surface

Any Hurwitz surface has triangulation as a quotient space of a , and the automorphisms of this triangulation coincide with Riemannian and algebraic automorphisms of the surface.

The Hurwitz surface is a compact Riemann surface that has exactly

84 ( g - 1)

automorphisms, where g is the genus of the surface. They are also called Hurwitz curves , understanding them as complex algebraic curves (complex dimension 1 corresponds to real dimension 2).

Named after the German mathematician Adolf Hurwitz .

Content

Properties

This number, 84 ( g - 1), by virtue of the Hurwitz automorphism theorem ^[1] , is maximal.

The Fuchsian group of the Hurwitz surface is a normal subgroup of finite index in the (ordinary) , and also is torsion-free. A finite quotient group is precisely a group of automorphisms.

Automorphisms of a complex algebraic curve are orientation-preserving automorphisms of the underlying real surface. If we consider also reversing the orientation of the isometry, then we get a twice as large group having the order of 168 ( g - 1), which is sometimes of interest.

Remarks

Here, the term “triangular group (2,3,7)” is most often understood as an incomplete triangular group Δ (2,3,7) ( the Coxeter group with the Schwartz triangle (2,3,7), or realized as a hyperbolic ), but rather an ordinary triangular group ( von Dick group ) D (2,3,7) of orientation-preserving mappings having index 2. The group of complex automorphisms is a factor group of an ordinary triangular group, while the isometry group (with a possible change in orientation) is a quotient group of a complete triangular group.

Examples

A Hurwitz surface of minimal genus is a genus 3, with a group of automorphisms PSL (2,7) (a projective special linear group) , having order 84 (3−1) = 168 = 2 ² • 3 • 7 and being simple group . The next admissible genus is seven and it has a MacBeath surface with a group of automorphisms PSL (2,8), which is a simple group of order 84 (7−1) = 504 = 2 ² • 3 ² • 7. If we also consider changing isometries, the order of the group will be 1008.

An interesting phenomenon is observed at the next possible value of the genus, namely at 14. Here there are three different Riemann surfaces with identical automorphism groups (of order 84 (14−1) = 1092 = 2 ² • 3 • 7 • 13). The explanation of this phenomenon is arithmetic. Namely, in a ring of integers of a suitable number field, the rational prime 13 is decomposed into the product of three different simple ideals ^[2] . defined by the triple of simple ideals give Fuchsian groups corresponding to the .

Notes

↑ Hurwitz, 1893 , p. 403–442.
↑ See the article “ ” for explanations.

Literature

N. Elkies . Shimura curve computations. Algorithmic number theory. - Berlin: Springer, 1998. - T. 1423. - (Lecture Notes in Computer Science).

A. Hurwitz. Über algebraische Gebilde mit Eindeutigen Transformationen in sich // Mathematische Annalen . - 1893. - T. 41 , no. 3 . - DOI : 10.1007 / BF01443420 .

M. Katz , M. Schaps, U. Vishne. Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups // J. Differential Geom. - 2007.- T. 76 , no. 3 . - S. 399-422 .

David Singerman, Robert I. Syddall. The Riemann Surface of a Uniform Dessin // Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry). - 2003. - T. 44 , no. 2 . - S. 413-430 .

[_770fe4c5659efb25-1] Hurwitz, 1893 , p. 403–442.

[2] See the article “ ” for explanations.