We study and classify all the conjugacy classes of the genus zero congruence subgroups of PSL2(ℝ) with no elliptic elements. We show that it suffices to classify those inside the modular group and determine them completely. We also discuss an application to modular curves.
"Torsion-free genus zero congruence subgroups of PSL2(ℝ)." Duke Math. J. 110 (2) 377 - 396, 1 November 2001. https://doi.org/10.1215/S0012-7094-01-11028-4