Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



The Deligne complex for the four-strand braid group

Author: Ruth Charney
Translated by:
Journal: Trans. Amer. Math. Soc. 356 (2004), 3881-3897
MSC (2000): Primary 20F36, 20F55, 52C35
Published electronically: December 15, 2003
MathSciNet review: 2058510
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper concerns the homotopy type of hyperplane arrangements associated to infinite Coxeter groups acting as reflection groups on $\mathbb C^n$. A long-standing conjecture states that the complement of such an arrangement should be aspherical. Some partial results on this conjecture were previously obtained by the author and M. Davis. In this paper, we extend those results to another class of Coxeter groups. The key technical result is that the spherical Deligne complex for the 4-strand braid group is CAT(1).

References [Enhancements On Off] (What's this?)

  • 1. M. Bridson and A. Haefliger.
    Metric Spaces of Non-positive Curvature.
    Springer-Verlag, Berlin, 1999. MR 2000k:53038
  • 2. R. Charney.
    Injectivity of the positive monoid for some infinite type Artin groups.
    In J. Cossey, C. F. Miller, W. D. Neumann, and M. Shapiro, editors, Geometric Group Theory Down Under, pages 103-118, Berlin, 1999. Walter de Gruyter. MR 2000h:20073
  • 3. R. Charney.
    The Tits conjecture for locally reducible Artin groups.
    International Journal of Algebra and Computation, 10:783-797, 2000. MR 2002d:20057
  • 4. R. Charney and M. W. Davis.
    Finite $K(\pi,1)$'s for Artin groups.
    In F. Quinn, editor, Prospects in Topology, number 138 in Annals of Math Studies, pages 110-124. Princeton University Press, 1995. MR 97a:57001
  • 5. R. Charney and M. W. Davis.
    The $K(\pi,1)$-problem for hyperplane complements associated to infinite reflection groups.
    Journal of the American Mathematical Society, 8(3):597-627, 1995. MR 95i:52011
  • 6. R. Charney and A. Lytchak.
    Metric characterizations of spherical and Euclidean buildings.
    Geometry and Topology, 5:521-550, 2001. MR 2002h:51008
  • 7. R. Charney and D. Peifer.
    The $K(\pi,1)$-conjecture for the affine braid groups.
    to appear in Commentarii Mathematici Helvetici.
  • 8. J. Crisp.
    Injective maps between Artin groups.
    In J. Cossey, C. F. Miller, W. D. Neumann, and M. Shapiro, editors, Geometric Group Theory Down Under, pages 119-137, Berlin, 1999. Walter de Gruyter. MR 2001b:20064
  • 9. J. Crisp.
    Symmetrical subgroups of Artin groups.
    Advances in Mathematics, 152:159-177, 2000. MR 2001c:20083
  • 10. M. W. Davis.
    Groups generated by reflections and aspherical manifolds not covered by Euclidean space.
    Annals of Mathematics, 117:293-324, 1983. MR 86d:57025
  • 11. P. Deligne.
    Les immeubles des groupes de tresses généralises.
    Inventiones Math., 17:273-302, 1972. MR 54:10659
  • 12. M. Elder and J. McCammond.
    Curvature testing in 3-dimensional metric polyhedral complexes.
    Experimental Mathematics, 11:143-158, 2002.
  • 13. E. Godelle.
    Normalisateurs et centralisateurs des sous-groupes paraboliques dans les groups d'Artin-Tits.
    Ph.D. thesis, Université de Picardie Jules Verne, 2001.
  • 14. M. Gromov.
    Hyperbolic groups.
    In Essays in Group Theory, number 8 in Math. Sci. Res. Inst. Publ., pages 75-264. Springer-Verlag, New York, 1987. MR 89e:20070
  • 15. G. Moussong.
    Hyperbolic Coxeter groups.
    Ph.D. thesis, Ohio State University, 1988.
  • 16. M. Salvetti.
    Topology of the complement of real hyperplanes in $\mathbb C^n$.
    Inventiones Mathematica, 88:603-618, 1987. MR 88k:32038
  • 17. H. van der Lek.
    Extended Artin groups.
    Proceedings of the Symposium in Pure Mathematics, 40:117-121, 1983. MR 85b:14005
  • 18. E. B. Vinberg.
    Discrete linear groups generated by reflections.
    Math. USSR Izvestija, 5(5):1083-1119, 1971. MR 46:1922

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 20F36, 20F55, 52C35

Retrieve articles in all journals with MSC (2000): 20F36, 20F55, 52C35

Additional Information

Ruth Charney
Affiliation: Department of Mathematics, The Ohio State University, 231 W. 18th Ave, Columbus, Ohio 43210
Address at time of publication: Department of Mathematics, Brandeis University, Waltham, Massachusetts 02254

Keywords: Artin groups, hyperplane arrangements
Received by editor(s): August 6, 2002
Received by editor(s) in revised form: May 1, 2003
Published electronically: December 15, 2003
Additional Notes: This work was partially supported by NSF grant DMS-0104026
Article copyright: © Copyright 2003 American Mathematical Society