In this question it is noted that $SL(2, \mathbb{R})/SL(2, \mathbb{Z})$ is homeomorphic to the trefoil complement in $S^3.$ Is there a similarly nice interpretation of $SL(2, \mathbb{R})/\Gamma(N)?$ for various $N?$
Unit tangent bundles of principal congruence orbifolds
Igor Rivin
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