- First Hurwitz triplet
In the mathematical theory of
Riemann surface s, thefirst Hurwitz triplet is a triple of distinctHurwitz surface s with the identical automorphism group of the lowest possible genus, namely 14 (genera 3 and 7 admit a unique Hurwitz surface, respectively theKlein quartic and theMacbeath surface ). 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 principalcongruence subgroup s defined by the triplet of primes produceFuchsian group s corresponding to the triplet of Riemann surfaces.Arithmetic construction
Let be the real subfield of where is a 7th-primitive
root of unity . Thering of integers of K is , where . Let be thequaternion algebra , or symbol algebra . Also Let and . Let . Then is a maximal order of (seeHurwitz quaternion order ), described explicitly byNoam Elkies [1] .In order to construct the first Hurwitz triplet, consider the prime decomposition of 13 in , namely
:,
where is invertible. Also consider the prime ideals generated by the non-invertible factors. The principal congruence subgroup defined by such a prime ideal "I" is by definition the group
:mod ,
namely, the group of elements of reduced norm 1 in equivalent to 1 modulo the ideal . The corresponding Fuchsian group is obtained as the image of the principal congruence subgroup under a representation to P
SL(2,R) .Each of the three Riemann surfaces in the first Hurwitz triplet can be formed as a
Fuchsian model , the quotient of thehyperbolic plane by one of these three Fuchsian groups.Bound for systolic length and the systolic ratio
The
Gauss-Bonnet theorem states that:
where is the
Euler characteristic of the surface and is theGaussian curvature . In the case we have: and
thus we obtain that the area of these surfaces is
:.
The lower bound on the systole as specified in [2] , namely
:
is 3.5187.
Some specific details about each of the surfaces are presented in the following tables (the number of systolic loops is taken from [3] ).The term Systolic Trace refers to the least reduced trace of an element in the corresponding subgroup . The systolic ratio is the ratio of the square of the systole to the area.
ee also
*
(2,3,7) triangle group References
* [1] Elkies, N.: The Klein quartic in number theory. The eightfold way, 51– 101, Math. Sci. Res. Inst. Publ. 35, Cambridge Univ. Press, Cambridge, 1999.
* [2] Katz, M.; Schaps, M.; Vishne, U.: Logarithmic growth of systole of arithmetic Riemann surfaces along congruence subgroups. J. Differential Geom. 76 (2007), 399-422. Available at arXiv:math.DG/0505007.
* [3] Vogeler, R.: On the geometry of Hurwitz surfaces. Thesis. Florida State University. 2003.
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