Systolic length of triangular modular curves

TL;DR

This paper introduces a method to compute upper bounds on the systolic length of certain Riemann surfaces using traces of Fuchsian group generators, with explicit computations and a growth rate analysis.

Contribution

It provides a novel approach to bounding systolic lengths of Riemann surfaces uniformized by congruence subgroups, including non-arithmetic cases, and demonstrates logarithmic growth with genus.

Findings

Systolic length bounds are derived from generator traces.

Explicit computations for various surfaces are presented.

Systolic length grows logarithmically with genus.

Abstract

We present a method for computing upper bounds on the systolic length of certain Riemann surfaces uniformized by congruence subgroups of hyperbolic triangle groups, admitting congruence Hurwitz curves as a special case. The uniformizing group is realized as a Fuchsian group and a convenient finite generating set is computed. The upper bound is derived from the traces of the generators. Some explicit computations, including ones for non-arithmetic surfaces, are given. We apply a result of Cosac and D\'{o}ria to show that the systolic length grows logarithmically with respect to the genus.