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
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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
- 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
- Stefan Papadima and Alexander I. Suciu, 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
- 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
- John Stallings, A finitely presented group whose 3-dimensional integral homology is not finitely generated, Amer. J. Math. 85 (1963), 541–543. MR 158917, DOI https://doi.org/10.2307/2373106
- D. Sullivan, 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
