With cedram.org
Table of contents for this issue | Previous article | Next article
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
Article PDF
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.


[1] A. Ash, D. Mumford, M. Rapoport & Y.-S. Tai, Smooth compactifications of locally symmetric varieties, Cambridge Mathematical Library, Cambridge University Press, 2010
[2] A. Beauville, Complex algebraic surfaces, London Mathematical Society Student Texts 34, Cambridge University Press, 1996
[3] D. Cartwright, V. Koziarz & S.-K. Yeung, “On the Cartwright–Steger surface”, https://arxiv.org/abs/1412.4137
[4] D. Cartwright & T. Steger, “Enumeration of the 50 fake projective planes”, C. R. Math. Acad. Sci. Paris 348 (2010) no. 1-2, p. 11-13  MR 2586735
[5] L. F. Di Cerbo, “Finite-volume complex-hyperbolic surfaces, their toroidal compactifications, and geometric applications”, Pacific J. Math. 255 (2012) no. 2, p. 305-315
[6] L. F. Di Cerbo & M. Stover, “Classification and arithmeticity of toroidal compactifications with $3\overline{c}_{2}=\overline{c}^{2}_{1}=3$”, https://arxiv.org/abs/1505.01414v2
[7] L. F. Di Cerbo & M. Stover, “Multiple realizations of varieties as ball quotient compactifications”, Michigan Math. J. 65 (2016) no. 2, p. 441-447
[8] W. Goldman, Complex hyperbolic geometry, Oxford Mathematical Monographs, Oxford University Press, 1999
[9] G. Harder, “A Gauss-Bonnet formula for discrete arithmetically defined groups”, Ann. Sci. École Norm. Sup. 4 (1971), p. 409-455
[10] F. Hirzebruch, “Chern numbers of algebraic surfaces: an example”, Math. Ann. 266 (1984) no. 3, p. 351-356
[11] M. Kapovich, “On normal subgroups of the fundamental groups of complex surfaces”, Preprint, 1998
[12] A. Kasparian & G. Sankaran, “Fundamental groups of toroidal compactifications”, https://arxiv.org/abs/1501.00053v2
[13] J. Kollár, Shafarevich maps and automorphic forms, Princeton University Press, 1995
[14] N. Mok, Projective algebraicity of minimal compactifications of complex-hyperbolic space forms of finite volume, Perspectives in analysis, geometry, and topology, Progr. Math. 296, Birkhäuser/Springer, 2012, p. 331–354
[15] A. Momot, “Irregular ball-quotient surfaces with non-positive Kodaira dimension”, Math. Res. Lett. 15 (2008) no. 6, p. 1187-1195
[16] D. Mumford, “Hirzebruch’s proportionality theorem in the noncompact case”, Invent. Math. 42 (1977), p. 239-272
[17] V. K. Murty & D. Ramakrishnan, The Albanese of unitary Shimura varieties, The zeta functions of Picard modular surfaces, Univ. Montréal, 1992, p. 445–464
[18] T. Napier & M. Ramachandran, “Hyperbolic Kähler manifolds and proper holomorphic mappings to Riemann surfaces”, Geom. Funct. Anal. 11 (2001) no. 2, p. 382-406
[19] M. V. Nori, “Zariski’s conjecture and related problems”, Ann. Sci. École Norm. Sup. 16 (1983) no. 2, p. 305-344
[20] Fernando Serrano, “Divisors of bielliptic surfaces and embeddings in ${\bf P}^4$”, Math. Z. 203 (1990) no. 3, p. 527-533  MR 1038716
[21] M. Stover, “Cusps and $b_{1}$ growth for ball quotients and maps onto $\mathbb{Z}$ with finitely generated kernel”, https://arxiv.org/abs/1506.06126v2
[22] William P. Thurston, Three-dimensional geometry and topology. Vol. 1, Princeton Mathematical Series 35, Princeton University Press, 1997, Edited by Silvio Levy
[23] G. Tian & S.-T. Yau, Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, Mathematical aspects of string theory (San Diego, Calif., 1986), Adv. Ser. Math. Phys. 1, World Sci. Publishing, 1987, p. 574–628
[24] S. Zucker, “$L_{2}$ cohomology of warped products and arithmetic groups”, Invent. Math. 70 (1982/83) no. 2, p. 169-218