Hostname: page-component-78c5997874-v9fdk Total loading time: 0 Render date: 2024-11-05T04:05:35.306Z Has data issue: false hasContentIssue false
  • English
  • Français

H 1–semistability for projective groups

Published online by Cambridge University Press:  31 May 2016

INDRANIL BISWAS  and
MAHAN MJ
INDRANIL BISWAS
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India. e-mails: [email protected]; [email protected]
MAHAN MJ
Affiliation:
School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India. e-mails: [email protected]; [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We initiate the study of the asymptotic topology of groups that can be realised as fundamental groups of smooth complex projective varieties with holomorphically convex universal covers (these are called here as holomorphically convex groups). We prove the H 1-semistability conjecture of Geoghegan for holomorphically convex groups. In view of a theorem of Eyssidieux, Katzarkov, Pantev and Ramachandran [EKPR], this implies that linear projective groups satisfy the H 1-semistability conjecture.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 162 , Issue 1 , January 2017 , pp. 89 - 100
Copyright
Copyright © Cambridge Philosophical Society 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[AN] Andreotti, A. and Narasimhan, R. A topological property of Runge pairs. Ann. of Math. 76 (1962), 499–509.Google Scholar
[BE] Bieri, R. and Eckmann, B. Groups with homological duality generalizing Poincaré duality. Invent. Math. 20 (1973), 103–124.CrossRefGoogle Scholar
[BMP] Biswas, I., Mj, M. and Pancholi, D. Homotopical height. Internat. J. Math. 25, no. 13, 1450123 (2014), (43 pages).Google Scholar
[Br] Brown, K. S. Cohomology of groups. Graduate Texts in Mathematics (Springer-Verlag, 1982).Google Scholar
[Dy] Dyer, M. N. On the second homotopy module of two-dimensional CW complexes. Proc. Amer. Math. Soc. 55 (1976), 400–404.CrossRefGoogle Scholar
[EKPR] Eyssidieux, P., Katzarkov, L., Pantev, T. and Ramachandran, M. Linear Shafarevich conjecture. Ann. of Math. 176 (2012), 1545–1581.CrossRefGoogle Scholar
[Ey] Eyssidieux, P. Sur la convexité holomorphe des revêtements linéaires réductifs d'une variété projective algébrique complexe. Invent. Math. 156 (2004), 503–564.Google Scholar
[Fa] Farrell, F. T. The second cohomology group of G with Z 2 G coefficients. Topology 13 (1974), 313–326.CrossRefGoogle Scholar
[Ge] Geoghegan, R. Topological Methods in Group Theory. Graduate Texts in Mathematics 243 (Springer, 2008).CrossRefGoogle Scholar
[GeMi1] Geoghegan, R. and Mihalik, M. L. Free abelian cohomology of groups and ends of universal covers. J. Pure Appl. Alg. 36 (1985), 123–137.Google Scholar
[GeMi2] Geoghegan, R. and Mihalik, M. L. A note on the vanishing of Hn (G, ℤG). J. Pure Appl. Alg. 39 (1986), 301–304.CrossRefGoogle Scholar
[GoMa] Goresky, M. and MacPherson, R. Stratified Morse theory. Ergebnisse der Mathematik und ihrer Grenzgebiete 14 (Springer-Verlag, Berlin, 1988).CrossRefGoogle Scholar
[Gui] Guilbault, C. R. Ends, shapes, and boundaries in manifold topology and geometric group theory. To appear in Springer Lecture Notes in Math. volume Topology and Geometric Group Theory (Proceedings of the OSU special year 2010-2011); arXiv:1210.6741Google Scholar
[Gur] Gurjar, R. V. Two remarks on the topology of projective surfaces. Math. Ann. 328 (2004), 701–706.CrossRefGoogle Scholar
[Hu] Hu, S. T. Homotopy theory. Pure and Appl. Math., vol. 8 (Academic Press, New York and London, 1959).Google Scholar
[Kl] Klingler, B. Kähler groups and duality. Preprint, arXiv:math 1005.2836.Google Scholar
[KR] Katzarkov, L. and Ramachandran, M. On the universal coverings of algebraic surfaces. Ann. Sci. École Norm. Sup. 31 (1998), 525–535.Google Scholar
[Mi1] Mihalik, M. L. Semistability at the end of a group extension. Trans. Amer. Math. Soc. 277 (1983), 307–321.Google Scholar
[Mi2] Mihalik, M. L. Semistability at ∞, ∞–ended groups and group cohomology. Trans. Amer. Math. Soc. 303 (1987), 479–485.Google Scholar
[Mi3] Mihalik, M. L. Semistability of Artin and Coxeter groups. J. Pure Appl. Alg. 111 (1996), 205–211.CrossRefGoogle Scholar
[MT1] Mihalik, M. L. and Tschantz, S. T. One relator groups are semistable at infinity. Topology 31 (1992), 801–804.Google Scholar
[MT2] Mihalik, M. L. and Tschantz, S. T. Semistability of amalgamated products and HNN-extensions. Mem. Amer. Math. Soc. 98 (1992), 471.Google Scholar
[Na] Narasimhan, R. On the homology groups of Stein spaces. Invent. Math. 2 (1967), 377–385.Google Scholar
[Re] Remmert, R. Sur les espaces analytiques holomorphiquement séparables et holomorphiquement convexes. Com. Ren. Acad. Sci. Paris 243 (1956), 118–121.Google Scholar