Volumes of 3-ball quotients as intersection numbers
HTML articles powered by AMS MathViewer
- by Martin Deraux PDF
- Trans. Amer. Math. Soc. 373 (2020), 343-383 Request permission
Abstract:
We give an explicit description of the 3-ball quotients constructed by Couwenberg-Heckman-Looijenga and deduce the value of their orbifold Euler characteristics. For each lattice, we also give a presentation in terms of generators and relations.References
- E. Artin, Braids and permutations, Ann. of Math. (2) 48 (1947), 643–649. MR 20989, DOI 10.2307/1969131
- David Bessis, Finite complex reflection arrangements are $K(\pi ,1)$, Ann. of Math. (2) 181 (2015), no. 3, 809–904. MR 3296817, DOI 10.4007/annals.2015.181.3.1
- David Bessis and Jean Michel, Explicit presentations for exceptional braid groups, Experiment. Math. 13 (2004), no. 3, 257–266. MR 2103323, DOI 10.1080/10586458.2004.10504537
- Armand Borel, Compact Clifford-Klein forms of symmetric spaces, Topology 2 (1963), 111–122. MR 146301, DOI 10.1016/0040-9383(63)90026-0
- Armand Borel and Harish-Chandra, Arithmetic subgroups of algebraic groups, Ann. of Math. (2) 75 (1962), 485–535. MR 147566, DOI 10.2307/1970210
- Michel Broué, Gunter Malle, and Raphaël Rouquier, Complex reflection groups, braid groups, Hecke algebras, J. Reine Angew. Math. 500 (1998), 127–190. MR 1637497
- Arjeh M. Cohen, Finite complex reflection groups, Ann. Sci. École Norm. Sup. (4) 9 (1976), no. 3, 379–436. MR 422448, DOI 10.24033/asens.1313
- Kevin Corlette, Archimedean superrigidity and hyperbolic geometry, Ann. of Math. (2) 135 (1992), no. 1, 165–182. MR 1147961, DOI 10.2307/2946567
- Wim Couwenberg, Gert Heckman, and Eduard Looijenga, Geometric structures on the complement of a projective arrangement, Publ. Math. Inst. Hautes Études Sci. 101 (2005), 69–161. MR 2217047, DOI 10.1007/s10240-005-0032-3
- P. Deligne and G. D. Mostow, Monodromy of hypergeometric functions and nonlattice integral monodromy, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 5–89. MR 849651, DOI 10.1007/BF02831622
- M. Deraux, A new non-arithmetic lattice in $\mathrm {PU}(3,1)$, to appear in Algebr. Geom. Topol., arXiv:1710.04463, 2017.
- M. Deraux, Non-arithmetic lattices and the Klein quartic, J. reine Angew. Math. (to appear), arXiv:1605.03846, 2016.
- Martin Deraux, Non-arithmetic ball quotients from a configuration of elliptic curves in an abelian surface, Comment. Math. Helv. 93 (2018), no. 3, 533–554. MR 3854901, DOI 10.4171/CMH/443
- M. Deraux, J. R. Parker, and J. Paupert, On commensurability classes of non-arithmetic complex hyperbolic lattices, preprint, arXiv:1611.00330, 2016.
- Martin Deraux, John R. Parker, and Julien Paupert, New non-arithmetic complex hyperbolic lattices, Invent. Math. 203 (2016), no. 3, 681–771. MR 3461365, DOI 10.1007/s00222-015-0600-1
- M. Gromov and I. Piatetski-Shapiro, Nonarithmetic groups in Lobachevsky spaces, Inst. Hautes Études Sci. Publ. Math. 66 (1988), 93–103. MR 932135
- Mikhail Gromov and Richard Schoen, Harmonic maps into singular spaces and $p$-adic superrigidity for lattices in groups of rank one, Inst. Hautes Études Sci. Publ. Math. 76 (1992), 165–246. MR 1215595, DOI 10.1007/BF02699433
- Friedrich Hirzebruch, Automorphe Formen und der Satz von Riemann-Roch, Symposium internacional de topología algebraica International symposium on algebraic topology, Universidad Nacional Autónoma de México and UNESCO, Mexico City, 1958, pp. 129–144 (German). MR 0103280
- János Kollár and Shigefumi Mori, Birational geometry of algebraic varieties, Cambridge Tracts in Mathematics, vol. 134, Cambridge University Press, Cambridge, 1998. With the collaboration of C. H. Clemens and A. Corti; Translated from the 1998 Japanese original. MR 1658959, DOI 10.1017/CBO9780511662560
- Vincent Koziarz and Duc-Manh Nguyen, Complex hyperbolic volume and intersection of boundary divisors in moduli spaces of pointed genus zero curves, Ann. Sci. Éc. Norm. Supér. (4) 51 (2018), no. 6, 1549–1597 (English, with English and French summaries). MR 3940904, DOI 10.24033/asens.2381
- G. A. Margulis, Discrete groups of motions of manifolds of nonpositive curvature, Proceedings of the International Congress of Mathematicians (Vancouver, B.C., 1974) Canad. Math. Congress, Montreal, Que., 1975, pp. 21–34 (Russian). MR 0492072
- Curtis T. McMullen, The Gauss-Bonnet theorem for cone manifolds and volumes of moduli spaces, Amer. J. Math. 139 (2017), no. 1, 261–291. MR 3619915, DOI 10.1353/ajm.2017.0005
- G. D. Mostow, Strong rigidity of locally symmetric spaces, Annals of Mathematics Studies, No. 78, Princeton University Press, Princeton, N.J.; University of Tokyo Press, Tokyo, 1973. MR 0385004
- G. D. Mostow, On a remarkable class of polyhedra in complex hyperbolic space, Pacific J. Math. 86 (1980), no. 1, 171–276. MR 586876, DOI 10.2140/pjm.1980.86.171
- G. D. Mostow, Generalized Picard lattices arising from half-integral conditions, Inst. Hautes Études Sci. Publ. Math. 63 (1986), 91–106. MR 849652, DOI 10.1007/BF02831623
- G. D. Mostow, Braids, hypergeometric functions, and lattices, Bull. Amer. Math. Soc. (N.S.) 16 (1987), no. 2, 225–246. MR 876959, DOI 10.1090/S0273-0979-1987-15510-8
- Gopal Prasad, Volumes of $S$-arithmetic quotients of semi-simple groups, Inst. Hautes Études Sci. Publ. Math. 69 (1989), 91–117. With an appendix by Moshe Jarden and the author. MR 1019962, DOI 10.1007/BF02698841
- R. Fox and L. Neuwirth, The braid groups, Math. Scand. 10 (1962), 119–126. MR 150755, DOI 10.7146/math.scand.a-10518
- Ichirô Satake, The Gauss-Bonnet theorem for $V$-manifolds, J. Math. Soc. Japan 9 (1957), 464–492. MR 95520, DOI 10.2969/jmsj/00940464
- G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. MR 59914, DOI 10.4153/cjm-1954-028-3
- Robert Steinberg, Invariants of finite reflection groups, Canadian J. Math. 12 (1960), 616–618. MR 117285, DOI 10.4153/CJM-1960-055-3
- William P. Thurston, Shapes of polyhedra and triangulations of the sphere, The Epstein birthday schrift, Geom. Topol. Monogr., vol. 1, Geom. Topol. Publ., Coventry, 1998, pp. 511–549. MR 1668340, DOI 10.2140/gtm.1998.1.511
Additional Information
- Martin Deraux
- Affiliation: Institut Fourier, Université Grenoble Alpes, 38610 Gières, France
- MR Author ID: 740008
- Received by editor(s): March 15, 2018
- Received by editor(s) in revised form: May 2, 2019, and May 25, 2019
- Published electronically: August 20, 2019
- © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 343-383
- MSC (2010): Primary 22E40; Secondary 32M15, 14N20
- DOI: https://doi.org/10.1090/tran/7925
- MathSciNet review: 4042878