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We study and classify all the conjugacy classes of the genus zero congruence subgroups of PSL 2 (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.doi:10.1215/s0012-7094-01-11028-4 fatcat:bvxtvdc2kjd25fukgaf64uul2e