# Finite index subgroups of the mapping class group with geometric meaning

I have got a question that is perhaps not precise in a mathematical sense. Is there a classification of all coverings of the moduli space of Riemann surfaces which are moduli spaces themselves, that is, they parametrize some geometric structure on a surface.

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You may be referring to the congruence subgroup problem: front.math.ucdavis.edu/0901.4663 –  Ian Agol Apr 18 '12 at 17:19