Hurwitz surface

From Wikipedia, the free encyclopedia

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.

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

The Hurwitz surface of least genus is the Klein quartic of genus 3. The next possible genus is 7, possessed by the Macbeath surface.

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. 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.

[edit] References

• Hurwitz, A. (1893). "Über algebraische Gebilde mit Eindeutigen Transformationen in sich". Mathematische Annalen 41: 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