For n ≥ 2, Deligne famously proved using the congruence subgroup property that the central extension of Sp(2n, Z) by Z determined by its preimage in the universal cover of Sp(2n, R) is not residually finite. On the other hand, the preimage of PSL(2, Z) in any connected cover of SL(2, R) is residually finite, and one can prove this very explicitly using nilpotent quotients. I will describe joint work with Domingo Toledo that develops methods, generalizing one interpretation of the argument for PSL(2, Z), to prove residual finiteness (in fact, linearity) of cyclic central extensions of fundamental groups of aspherical manifolds with residually finite fundamental group. I will then describe how this generalization applies to prove residual finiteness of cyclic central extensions of certain arithmetic lattices in PU(n, 1).