An infinite family of perfect, non-extremal Riemann surfaces is constructed, the first examples of this type of surfaces. The examples are based on normal subgroups of the modular group $\PSL(2,{\sf Z})$ of level $6$. They provide non-Euclidean analogues to the existence of perfect, non-extremal positive definite quadratic forms. The analogy uses the function {\it syst\/} which associates to every Riemann surface $M$ the length of a systole, which is a shortest closed geodesic of $M$.

MSC Classifications:

11H99, 11F06, 30F45 show english descriptions None of the above, but in this section
Structure of modular groups and generalizations; arithmetic groups [See also 20H05, 20H10, 22E40]
Conformal metrics (hyperbolic, Poincare, distance functions) 11H99 - None of the above, but in this section
11F06 - Structure of modular groups and generalizations; arithmetic groups [See also 20H05, 20H10, 22E40]
30F45 - Conformal metrics (hyperbolic, Poincare, distance functions)

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