Quasi-Kähler Bestvina-Brady groups

Authors:
Alexandru Dimca, Stefan Papadima and Alexander I. Suciu

Journal:
J. Algebraic Geom. **17** (2008), 185-197

DOI:
https://doi.org/10.1090/S1056-3911-07-00463-8

Published electronically:
June 27, 2007

MathSciNet review:
2357684

Full-text PDF

Abstract |
References |
Additional Information

Abstract: A finite simple graph $\Gamma$ determines a right-angled Artin group $G_\Gamma$, with one generator for each vertex $v$, and with one commutator relation $vw=wv$ 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 $v$ to $1$. 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.

References
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*Formality, Alexander invariants, and a question of Serre*, preprint arXiv:math.AT/0512480.
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*Non-finiteness properties of fundamental groups of smooth projective varieties*, preprint arXiv:math.AG/0609456.
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*Isomorphisms of graph groups*, Proc. Amer. Math. Soc. **100** (1987), no. 3, 407–408. MR **88e:20033**
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*Chen Lie algebras*, Internat. Math. Res. Notices **2004** (2004), no. 21, 1057–1086. MR **2004m:17043**
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*Algebraic invariants for right-angled Artin groups*, Math. Annalen **334** (2006), no. 3, 533–555. MR **2006k:20078**
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*When does the associated graded Lie algebra of an arrangement group decompose?*, Comment. Math. Helv. **81** (2006), no. 4, 859–875. MR **2271225**, DOI https://doi.org/10.4171/CMH/77
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*Rational homotopy theory*, Ann. of Math. **90** (1969), 205–295. MR **41:2678**
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*Infinitesimal computations in topology*, Inst. Hautes Études Sci. Publ. Math. **47** (1977), 269–331. MR **58:31119**

References
- 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**
- G. Baumslag,
*Lecture notes on nilpotent groups*, CBMS Regional Conference Series in Math., no. 2, Amer. Math. Soc., Providence, 1971. MR **44:315**
- M. Bestvina, N. Brady,
*Morse theory and finiteness properties of groups*, Invent. Math. **129** (1997), no. 3, 445–470. MR **98i:20039**
- 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**
- S.A. Broughton,
*Milnor numbers and the topology of polynomial hypersurfaces*, Invent. Math. **92** (1988), 217–241. MR **89h:32025**
- K.S. Brown,
*Cohomology of groups*, Grad. Texts in Math., vol. 87, Springer-Verlag, New York-Berlin, 1982. MR **83k:20002**
- 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**
- 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**
- W. Dicks, I. Leary,
*Presentations for subgroups of Artin groups*, Proc. Amer. Math. Soc. **127** (1999), no. 2, 343–348. MR **99c:20050**
- 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**
- A. Dimca, S. Papadima, A. Suciu,
*Formality, Alexander invariants, and a question of Serre*, preprint arXiv:math.AT/0512480.
- A. Dimca, S. Papadima, A. Suciu,
*Non-finiteness properties of fundamental groups of smooth projective varieties*, preprint arXiv:math.AG/0609456.
- C. Droms,
*Isomorphisms of graph groups*, Proc. Amer. Math. Soc. **100** (1987), no. 3, 407–408. MR **88e:20033**
- 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**
- 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**
- 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**
- K.H. Kim, L. Makar-Limanov, J. Neggers, F.W. Roush,
*Graph algebras*, J. Algebra **64** (1980), no. 1, 46–51. MR **82e:05114a**
- J. Kollár,
*Shafarevich maps and automorphic forms*, Princeton Univ. Press, Princeton, NJ, 1995. MR **96i:14016**
- S. Papadima, A. Suciu,
*Chen Lie algebras*, Internat. Math. Res. Notices **2004** (2004), no. 21, 1057–1086. MR **2004m:17043**
- S. Papadima, A. Suciu,
*Algebraic invariants for right-angled Artin groups*, Math. Annalen **334** (2006), no. 3, 533–555. MR **2006k:20078**
- 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**
- S. Papadima, A. Suciu,
*Algebraic invariants for Bestvina-Brady groups*, to appear in J. London Math. Soc., available at arXiv:math.GR/0603240.
- D. Quillen,
*Rational homotopy theory*, Ann. of Math. **90** (1969), 205–295. MR **41:2678**
- J. Stallings,
*A finitely presented group whose $3$-dimensional integral homology is not finitely generated*, Amer. J. Math. **85** (1963), 541–543. MR **0158917**
- D. Sullivan,
*Infinitesimal computations in topology*, Inst. Hautes Études Sci. Publ. Math. **47** (1977), 269–331. MR **58:31119**

Additional Information

**Alexandru Dimca**

Affiliation:
Laboratoire J. A. Dieudonné, UMR du CNRS 6621, Université de Nice–Sophia Antipolis, Parc Valrose, 06108 Nice Cedex 02, France

MR Author ID:
58125

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

MR Author ID:
168600

ORCID:
0000-0002-5060-7754

Email:
a.suciu@neu.edu

Received by editor(s):
March 22, 2006

Received by editor(s) in revised form:
July 28, 2006

Published electronically:
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