Bielliptic ball quotient compactifications and lattices in PU(2,1) with finitely generated commutator subgroup
Annales de l'Institut Fourier, Volume 67 (2017) no. 1, p. 315-328

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 8 3π 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 PU(2,1), and the first infinite tower, with this property.

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 8 3π 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 PU(2,1) (ainsi que la première tour infinie) avec cette propriété.

Received : 2015-12-20
Revised : 2016-03-25
Accepted : 2016-06-14
Published online : 2017-01-10
Classification:  32Q45,  14M27,  57M50
Keywords: Ball quotients and their compactifications, volumes of complex hyperbolic manifolds
     author = {Di Cerbo, Luca F. and Stover, Matthew},
     title = {Bielliptic ball quotient compactifications and lattices in $\text{PU}(2, 1)$ with finitely generated commutator subgroup},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {1},
     year = {2017},
     pages = {315-328},
     doi = {10.5802/aif.3083},
     language = {en},
     url = {}
Di Cerbo, Luca F.; Stover, Matthew. Bielliptic ball quotient compactifications and lattices in $\text{PU}(2, 1)$ with finitely generated commutator subgroup. Annales de l'Institut Fourier, Volume 67 (2017) no. 1, pp. 315-328. doi : 10.5802/aif.3083.

[1] Ash, A.; Mumford, D.; Rapoport, M.; Tai, Y.-S. Smooth compactifications of locally symmetric varieties, Cambridge University Press, Cambridge Mathematical Library (2010)

[2] Beauville, A. Complex algebraic surfaces, Cambridge University Press, London Mathematical Society Student Texts, Tome 34 (1996)

[3] Cartwright, D.; Koziarz, V.; Yeung, S.-K. On the Cartwright–Steger surface ( )

[4] Cartwright, D.; Steger, T. Enumeration of the 50 fake projective planes, C. R. Math. Acad. Sci. Paris, Tome 348 (2010) no. 1-2, pp. 11-13

[5] Di Cerbo, L. F. Finite-volume complex-hyperbolic surfaces, their toroidal compactifications, and geometric applications, Pacific J. Math., Tome 255 (2012) no. 2, pp. 305-315

[6] Di Cerbo, L. F.; Stover, M. Classification and arithmeticity of toroidal compactifications with 3c ¯ 2 =c ¯ 1 2 =3 ( )

[7] Di Cerbo, L. F.; Stover, M. Multiple realizations of varieties as ball quotient compactifications, Michigan Math. J., Tome 65 (2016) no. 2, pp. 441-447

[8] Goldman, W. Complex hyperbolic geometry, Oxford University Press, Oxford Mathematical Monographs (1999)

[9] Harder, G. A Gauss-Bonnet formula for discrete arithmetically defined groups, Ann. Sci. École Norm. Sup., Tome 4 (1971), pp. 409-455

[10] Hirzebruch, F. Chern numbers of algebraic surfaces: an example, Math. Ann., Tome 266 (1984) no. 3, pp. 351-356

[11] Kapovich, M. On normal subgroups of the fundamental groups of complex surfaces (Preprint, 1998)

[12] Kasparian, A.; Sankaran, G. Fundamental groups of toroidal compactifications ( )

[13] Kollár, J. Shafarevich maps and automorphic forms, Princeton University Press (1995)

[14] Mok, N. Projective algebraicity of minimal compactifications of complex-hyperbolic space forms of finite volume, Perspectives in analysis, geometry, and topology, Birkhäuser/Springer (Progr. Math.) Tome 296 (2012), pp. 331-354

[15] Momot, A. Irregular ball-quotient surfaces with non-positive Kodaira dimension, Math. Res. Lett., Tome 15 (2008) no. 6, pp. 1187-1195

[16] Mumford, D. Hirzebruch’s proportionality theorem in the noncompact case, Invent. Math., Tome 42 (1977), pp. 239-272

[17] Murty, V. K.; Ramakrishnan, D. The Albanese of unitary Shimura varieties, The zeta functions of Picard modular surfaces, Univ. Montréal (1992), pp. 445-464

[18] Napier, T.; Ramachandran, M. Hyperbolic Kähler manifolds and proper holomorphic mappings to Riemann surfaces, Geom. Funct. Anal., Tome 11 (2001) no. 2, pp. 382-406

[19] Nori, M. V. Zariski’s conjecture and related problems, Ann. Sci. École Norm. Sup., Tome 16 (1983) no. 2, pp. 305-344

[20] Serrano, Fernando Divisors of bielliptic surfaces and embeddings in P 4 , Math. Z., Tome 203 (1990) no. 3, pp. 527-533

[21] Stover, M. Cusps and b 1 growth for ball quotients and maps onto with finitely generated kernel ( )

[22] Thurston, William P. Three-dimensional geometry and topology. Vol. 1, Princeton University Press, Princeton Mathematical Series, Tome 35 (1997) (Edited by Silvio Levy)

[23] Tian, G.; Yau, S.-T. Existence of Kähler-Einstein metrics on complete Kähler manifolds and their applications to algebraic geometry, Mathematical aspects of string theory (San Diego, Calif., 1986), World Sci. Publishing (Adv. Ser. Math. Phys.) Tome 1 (1987), pp. 574-628

[24] Zucker, S. L 2 cohomology of warped products and arithmetic groups, Invent. Math., Tome 70 (1982/83) no. 2, pp. 169-218