I will relate two notorious open questions in low-dimensional topology.  The first asks whether every hyperbolic group is residually finite. The second, the  congruence subgroup property, relates the finite-index subgroups of mapping class groups of surfaces to the topology of the underlying surface. I will explain why, if every hyperbolic group is residually finite, then mapping class groups enjoy the congruence subgroup property. If there’s time, I may give some further applications to the question of whether hyperbolic 3-manifolds are determined by the finite quotients of their fundamental groups.