## The Deligne complex for the four-strand braid group

HTML articles powered by AMS MathViewer

- by Ruth Charney PDF
- Trans. Amer. Math. Soc.
**356**(2004), 3881-3897 Request permission

## 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

- Martin R. Bridson and André Haefliger,
*Metric spaces of non-positive curvature*, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319, Springer-Verlag, Berlin, 1999. MR**1744486**, DOI 10.1007/978-3-662-12494-9 - Ruth Charney,
*Injectivity of the positive monoid for some infinite type Artin groups*, Geometric group theory down under (Canberra, 1996) de Gruyter, Berlin, 1999, pp. 103–118. MR**1714841** - Ruth Charney,
*The Tits conjecture for locally reducible Artin groups*, Internat. J. Algebra Comput.**10**(2000), no. 6, 783–797. MR**1809385**, DOI 10.1142/S0218196700000479 - Ruth Charney and Michael W. Davis,
*Finite $K(\pi , 1)$s for Artin groups*, Prospects in topology (Princeton, NJ, 1994) Ann. of Math. Stud., vol. 138, Princeton Univ. Press, Princeton, NJ, 1995, pp. 110–124. MR**1368655** - Ruth Charney and Michael W. Davis,
*The $K(\pi ,1)$-problem for hyperplane complements associated to infinite reflection groups*, J. Amer. Math. Soc.**8**(1995), no. 3, 597–627. MR**1303028**, DOI 10.1090/S0894-0347-1995-1303028-9 - Ruth Charney and Alexander Lytchak,
*Metric characterizations of spherical and Euclidean buildings*, Geom. Topol.**5**(2001), 521–550. MR**1833752**, DOI 10.2140/gt.2001.5.521 - R. Charney and D. Peifer. The $K(\pi ,1)$-conjecture for the affine braid groups. to appear in Commentarii Mathematici Helvetici.
- John Crisp,
*Injective maps between Artin groups*, Geometric group theory down under (Canberra, 1996) de Gruyter, Berlin, 1999, pp. 119–137. MR**1714842** - John Crisp,
*Symmetrical subgroups of Artin groups*, Adv. Math.**152**(2000), no. 1, 159–177. MR**1762124**, DOI 10.1006/aima.1999.1895 - Michael W. Davis,
*Groups generated by reflections and aspherical manifolds not covered by Euclidean space*, Ann. of Math. (2)**117**(1983), no. 2, 293–324. MR**690848**, DOI 10.2307/2007079 - Pierre Deligne,
*Les immeubles des groupes de tresses généralisés*, Invent. Math.**17**(1972), 273–302 (French). MR**422673**, DOI 10.1007/BF01406236 - M. Elder and J. McCammond. Curvature testing in 3-dimensional metric polyhedral complexes.
*Experimental Mathematics*, 11:143–158, 2002. - E. Godelle.
*Normalisateurs et centralisateurs des sous-groupes paraboliques dans les groups d’Artin-Tits*. Ph.D. thesis, Université de Picardie Jules Verne, 2001. - M. Gromov,
*Hyperbolic groups*, Essays in group theory, Math. Sci. Res. Inst. Publ., vol. 8, Springer, New York, 1987, pp. 75–263. MR**919829**, DOI 10.1007/978-1-4613-9586-7_{3} - G. Moussong.
*Hyperbolic Coxeter groups*. Ph.D. thesis, Ohio State University, 1988. - M. Salvetti,
*Topology of the complement of real hyperplanes in $\textbf {C}^N$*, Invent. Math.**88**(1987), no. 3, 603–618. MR**884802**, DOI 10.1007/BF01391833 - Harm van der Lek,
*Extended Artin groups*, Singularities, Part 2 (Arcata, Calif., 1981) Proc. Sympos. Pure Math., vol. 40, Amer. Math. Soc., Providence, RI, 1983, pp. 117–121. MR**713240** - È. B. Vinberg,
*Discrete linear groups that are generated by reflections*, Izv. Akad. Nauk SSSR Ser. Mat.**35**(1971), 1072–1112 (Russian). MR**0302779**

## 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
- MR Author ID: 47560
- Email: charney@math.ohio-state.edu, charney@brandeis.edu
- 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
- © Copyright 2003 American Mathematical Society
- 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
- MathSciNet review: 2058510