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We construct two infinite families of ball quotient compactificationsbirational to bielliptic surfaces. For each family, the volume spectrum of theassociated noncompact finite volume ball quotient surfaces is the set of allpositive integral multiples of $\frac{8}{3}\pi^{2}$, i.e., they attain allpossible volumes of complex hyperbolic $2$-manifolds. The surfaces in one ofthe two families have all $2$-cusps, so that we can saturate the entire volumespectrum with $2$-cusped manifolds. Finally, we show that the associated neatlattices have infinite abelianization and finitely generated commutatorsubgroup. These appear to be the first known nonuniform lattices in$\mathrm{PU}(2,1)$, and the first infinite tower, with this property.

Bielliptic ball quotient compactifications and lattices in PU(2, 1) with finitely generated commutator subgroup