# 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 virtue of Hurwitz's theorem on automorphisms (Hurwitz 1893). They are also referred to as **Hurwitz curves**, interpreting them as complex algebraic curves (complex dimension 1 = real dimension 2).

The Fuchsian group of a Hurwitz surface is a finite index torsionfree normal subgroup of the (ordinary) (2,3,7) triangle group. The finite quotient group is precisely the automorphism group.

Automorphisms of complex algebraic curves are *orientation-preserving* 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 with Schwarz triangle (2,3,7) or a realization as a hyperbolic reflection group), but rather to the *ordinary* triangle group (the von Dyck group) *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.

## Examples

The Hurwitz surface of least genus is the Klein quartic of genus 3, with automorphism group the projective special linear group PSL(2,7), of order 84(3−1) = 168 = 2^{2}·3·7, which is a simple group; (or order 336 if one allows orientation-reversing isometries). The next possible genus is 7, possessed by the Macbeath surface, with automorphism group PSL(2,8), which is the simple group of order 84(7−1) = 504 = 2^{2}·3^{2}·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 = 2^{2}·3·7·13). The explanation for this phenomenon is arithmetic. Namely, in the ring of integers of the appropriate number field, the rational prime 13 splits as a product of three distinct prime ideals. The principal congruence subgroups defined by the triplet of primes produce Fuchsian groups corresponding to the first Hurwitz triplet.

## See also

## References

- Elkies, N.: Shimura curve computations.
*Algorithmic number theory*(Portland, OR, 1998), 1–47, Lecture Notes in Computer Science, 1423, Springer, Berlin, 1998. See arXiv:math.NT/0005160 - Hurwitz, A. (1893). "Über algebraische Gebilde mit Eindeutigen Transformationen in sich".
*Mathematische Annalen*.**41**(3): 403–442. doi:10.1007/BF01443420. - Katz, M.; Schaps, M.; Vishne, U.: Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups. J. Differential Geom. 76 (2007), no. 3, 399-422. Available at arXiv:math.DG/0505007
- Singerman, David; Syddall, Robert I. (2003). "The Riemann Surface of a Uniform Dessin".
*Beiträge zur Algebra und Geometrie (Contributions to Algebra and Geometry)*. 44 (2): 413–430, PDF External link in`|journal=`

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