Journal of Algebraic Geometry

Author(s): Alexandru Dimca; Stefan Papadima; Alexander I. Suciu
Journal: J. Algebraic Geom. 17 (2008), 185-197.
Posted: June 27, 2007
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Abstract: A finite simple graph $\Gamma$ determines a right-angled Artin group $G_\Gamma$ , with one generator for each vertex

, and with one commutator relation

for each pair of vertices joined by an edge. The Bestvina-Brady group $N_\Gamma$ is the kernel of the projection $G_\Gamma\to \mathbb{Z}$ , which sends each generator

. We establish precisely which graphs $\Gamma$ give rise to quasi-Kähler (respectively, Kähler) groups $N_\Gamma$ . This yields examples of quasi-projective groups which are not commensurable (up to finite kernels) to the fundamental group of any aspherical, quasi-projective variety.

1.: J. Amorós, M. Burger, K. Corlette, D. Kotschick, D. Toledo, Fundamental groups of compact Kähler manifolds, Math. Surveys Monogr., vol. 44, Amer. Math. Soc., Providence, RI, 1996. MR 97d:32037
2.: G. Baumslag, Lecture notes on nilpotent groups, CBMS Regional Conference Series in Math., no. 2, Amer. Math. Soc., Providence, 1971. MR 44:315
3.: M. Bestvina, N. Brady, Morse theory and finiteness properties of groups, Invent. Math. 129 (1997), no. 3, 445-470. MR 98i:20039
4.: R. Bieri, Homological dimension of discrete groups, Second edition, Queen Mary Coll. Math. Notes, Queen Mary College, Dept. Pure Math., London, 1981. MR 84h:20047
5.: S.A. Broughton, Milnor numbers and the topology of polynomial hypersurfaces, Invent. Math. 92 (1988), 217-241. MR 89h:32025
6.: K.S. Brown, Cohomology of groups, Grad. Texts in Math., vol. 87, Springer-Verlag, New York-Berlin, 1982. MR 83k:20002
7.: A.D.R. Choudary, A. Dimca, S. Papadima, Some analogs of Zariski's theorem on nodal line arrangements, Algebr. Geom. Topol. 5 (2005), 691-711. MR 2006f:32038
8.: P. Deligne, P. Griffiths, J. Morgan, D. Sullivan, Real homotopy theory of Kähler manifolds, Invent. Math. 29 (1975), no. 3, 245-274. MR 52:3584
9.: W. Dicks, I. Leary, Presentations for subgroups of Artin groups, Proc. Amer. Math. Soc. 127 (1999), no. 2, 343-348. MR 99c:20050
10.: A. Dimca, Hyperplane arrangements, M-tame polynomials and twisted cohomology, in: Commutative Algebra, Singularities and Computer Algebra (Sinaia 2002), Kluwer, NATO Science Series, vol. 115, 2003, pp. 113-126. MR 2005b:32062
11.: A. Dimca, S. Papadima, A. Suciu, Formality, Alexander invariants, and a question of Serre, preprint arXiv:math.AT/0512480.
12.: A. Dimca, S. Papadima, A. Suciu, Non-finiteness properties of fundamental groups of smooth projective varieties, preprint arXiv:math.AG/0609456.
13.: C. Droms, Isomorphisms of graph groups, Proc. Amer. Math. Soc. 100 (1987), no. 3, 407-408. MR 88e:20033
14.: P. Hilton, G. Mislin, J. Roitberg, Localization of nilpotent groups and spaces, North-Holland Math. Studies, vol. 15, North-Holland, Amsterdam, 1975.MR 57:17635
15.: F.E.A. Johnson, E. Rees, On the fundamental group of a complex algebraic manifold, Bull. London Math. Soc. 19 (1987), no. 5, 463-466. MR 89i:32016
16.: M. Kapovich, J. Millson, On representation varieties of Artin groups, projective arrangements and the fundamental groups of smooth complex algebraic varieties, Inst. Hautes Études Sci. Publ. Math. 88 (1998), 5-95. MR 2001d:14024
17.: K.H. Kim, L. Makar-Limanov, J. Neggers, F.W. Roush, Graph algebras, J. Algebra 64 (1980), no. 1, 46-51. MR 82e:05114a
18.: J. Kollár, Shafarevich maps and automorphic forms, Princeton Univ. Press, Princeton, NJ, 1995. MR 96i:14016
19.: S. Papadima, A. Suciu, Chen Lie algebras, Internat. Math. Res. Notices 2004 (2004), no. 21, 1057-1086. MR 2004m:17043
20.: S. Papadima, A. Suciu, Algebraic invariants for right-angled Artin groups, Math. Annalen 334 (2006), no. 3, 533-555. MR 2006k:20078
21.: S. Papadima, A. Suciu, When does the associated graded Lie algebra of an arrangement group decompose?, Commentarii Math. Helvetici 81 (2006), no. 4, 859-875. MR 2271225
22.: S. Papadima, A. Suciu, Algebraic invariants for Bestvina-Brady groups, to appear in J. London Math. Soc., available at arXiv:math.GR/0603240.
23.: D. Quillen, Rational homotopy theory, Ann. of Math. 90 (1969), 205-295. MR 41:2678
24.: J. Stallings, A finitely presented group whose -dimensional integral homology is not finitely generated, Amer. J. Math. 85 (1963), 541-543. MR 0158917
25.: D. Sullivan, Infinitesimal computations in topology, Inst. Hautes Études Sci. Publ. Math. 47 (1977), 269-331. MR 58:31119

Alexandru Dimca
Affiliation: Laboratoire J. A. Dieudonné, UMR du CNRS 6621, Université de Nice--Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France
Email: dimca@math.unice.fr

Stefan Papadima
Affiliation: Instititue of Mathematics Simion Stoilow, Romanian Academy, P.O. Box 1-764, RO-014700 Bucharest, Romania
Email: Stefan.Papadima@imar.ro

Alexander I. Suciu
Affiliation: Department of Mathematics, Northeastern University, Boston, Massachusetts 02115
Email: a.suciu@neu.edu

PII: S 1056-3911(07)00463-8
Received by editor(s): March 22, 2006
Received by editor(s) in revised form: July 28, 2006
Posted: June 27, 2007
Additional Notes: The second author was partially supported by CERES grant 4-147/12.11.2004 of the Romanian Ministry of Education and Research. The third author was partially supported by NSF grant DMS-0311142

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