Bielliptic ball quotient compactifications and lattices in $\text{PU}(2,1)$ with finitely generated commutator subgroup

Di Cerbo, Luca F.; Stover, Matthew

Annales de l'Institut Fourier, Tome 67 (2017) no. 1, pp. 315-328.

Nous construisons deux familles infinies de quotients de la boule non-compacts de volume fini qui admettent une compactification birationnelle à une surface bi-elliptique. Pour chaque famille, l’ensemble des volumes consiste en tous les multiples entiers positifs de $\frac{8}{3}{\pi }^2$, donc il réalise tous les volumes possibles pour une variété hyperbolique complexe de dimension $2$. Dans une des deux familles, toutes les surfaces ont exactement deux pointes, donc nous pouvons réaliser tout le spectre des volumes par des surfaces à deux pointes. Enfin, nous montrons que les réseaux associés (sans torsion, y compris á l’infini) ont un abélianisé infini, et un groupe dérivé de type fini. Ceux-ci semblent être les premiers réseaux non-uniformes connus dans $\mathrm{PU}(2,1)$ (ainsi que la première tour infinie) avec cette propriété.

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.

Classification : 32Q45, 14M27, 57M50
Keywords: Ball quotients and their compactifications, volumes of complex hyperbolic manifolds
Mot clés : Quotients de la boule et leurs compactifications, volumes des variétés hyperboliques complexes

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