  We establish an analog of a theorem of Stallings which asserts the
homomorphisms between the universal nilpotent quotients induced by a
homomorphism $G \to H$ of groups are isomorphisms provided a pair of
homological conditions are satisfied. Our analogy does not have a homomorphism
between $G$ and $H$ but instead we have a diamond of groups containing $G,H$
that satisfies a similar homological condition. We derive a few applications of
this result. First, we show that there exist non-isomorphic number fields whose
absolute Galois groups have isomorphic universal nilpotent quotients. We show
that there exists pairs of non-isometric hyperbolic $n$-manifolds whose
fundamental groups are residually nilpotent and have isomorphic universal
nilpotent quotients. These are the first examples of residually nilpotent
Kleinian groups with arbitrarily large nilpotent genus. Complex hyperbolic
2-manifold examples are given which also supply examples of surfaces whose
geometric fundamental groups have the isomorphic universal nilpotent quotients
and the isomorphisms are compatible with the outer Galois actions. The case of
Riemann surfaces provides examples of curves where the same implication holds.
In fact, for every $g \geq 2$, we show all but finitely many curves in
$\mathcal{M}_g$ have finite covers where this happens.
