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

DOI:
https://doi.org/10.1090/S0002-9947-03-03425-1

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 . 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).

**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 '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 -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 -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 .*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**

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

Email:
charney@math.ohio-state.edu, charney@brandeis.edu

DOI:
https://doi.org/10.1090/S0002-9947-03-03425-1

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