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

TL;DR

This paper constructs new examples of complex hyperbolic 2-manifolds with specific volume spectra and properties of their lattices, advancing understanding of nonuniform lattices in PU(2,1).

Contribution

It introduces two infinite families of ball quotient compactifications birational to bielliptic surfaces with novel lattice properties.

Findings

Volume spectrum includes all positive multiples of 8/3 π^2.

Surfaces in one family have all 2-cusps, saturating the volume spectrum.

Associated lattices have infinite abelianization and finitely generated commutator subgroup.

Abstract

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