For $n \ge 2$, Deligne famously proved using the congruence subgroup property that the central extension of $\mathrm{Sp}(2n,\mathbb{Z})$ by $\mathbb{Z}$ determined by its preimage in the universal cover of $\mathrm{Sp}(2n,\mathbb{R})$ is not residually finite. On the other hand, the preimage of $\mathrm{PSL}(2,\mathbb{Z})$ in any connected cover of $\mathrm{SL}(2,\mathbb{R})$ is residually finite, and one can prove this very explicitly using nilpotent quotients. These quotients have many interpretations that manifest the modular group's special place at the intersection of low-dimensional topology, geometric group theory, number theory, and complex analysis. I will describe joint work with Domingo Toledo that develops methods, generalizing one interpretation of the argument for $\mathrm{PSL}(2,\mathbb{Z})$, 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 $\mathrm{PU}(n,1)$ and raise some open questions.

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