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Luca F. Di Cerbo; Matthew Stover
Bielliptic ball quotient compactifications and lattices in $\text{PU}(2, 1)$ with finitely generated commutator subgroup
(Compactifications bi-elliptiques de quotients de la boule et réseaux dans $\mathrm{PU}(2,1)$ dont le sous-groupe dérivé est de type fini)
Annales de l'institut Fourier, 67 no. 1 (2017), p. 315-328, doi: 10.5802/aif.3083
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Class. Math.: 32Q45, 14M27, 57M50
Keywords: Ball quotients and their compactifications, volumes of complex hyperbolic manifolds

Résumé - 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 all have $2$-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.

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