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, respectively the Klein quartic and the Macbeath 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 principal congruence subgroups defined by the triplet of primes produce Fuchsian groups corresponding to the triplet of Riemann surfaces.
Contents
Arithmetic construction
Let
In order to construct the first Hurwitz triplet, consider the prime decomposition of 13 in
where
namely, the group of elements of reduced norm 1 in
Each of the three Riemann surfaces in the first Hurwitz triplet can be formed as a Fuchsian model, the quotient of the hyperbolic plane by one of these three Fuchsian groups.
Bound for systolic length and the systolic ratio
The Gauss–Bonnet theorem states that
where
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