Geometriae Dedicata

, Volume 182, Issue 1, pp 263–286 | Cite as

The 6-strand braid group is CAT(0)

  • Thomas Haettel
  • Dawid Kielak
  • Petra SchwerEmail author
Original Paper


We show that braid groups with at most 6 strands are CAT(0) using the close connection between these groups, the associated non-crossing partition complexes, and the embeddability of their diagonal links into spherical buildings of type A. Furthermore, we prove that the orthoscheme complex of any bounded graded modular complemented lattice is CAT(0), giving a partial answer to a conjecture of Brady and McCammond.


Braid groups CAT(0) Non-crossing partitions Buildings Orthoscheme complexes Modular lattice 

Mathematics Subject Classification

20F65 20F36 



We would like to thank Piotr Przytycki for introducing us to the problem, and all the help provided during our work. We would also like to thank Brian Bowditch who helped us access his ‘Notes on locally CAT(1) spaces’ [3]. Finally, we would like to thank Ursula Hamenstädt for useful conversations, and the referee, for pointing out numerous ways of improving the presentation of this paper.


  1. 1.
    Abramenko, P., Brown, K.S.: Buildings. Theory and Applications, Graduate Texts in Mathematics, vol. 248. Springer, New York (2008)zbMATHGoogle Scholar
  2. 2.
    Bell, R.W.: Three-dimensional FC Artin groups are CAT(0). Geom. Dedicata 113, 21–53 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bowditch, B.H.: Notes on locally \({\rm CAT}(1)\) spaces. In: Geometric group theory (Columbus, OH, 1992), Ohio State University Mathematical Research Institute Publications, vol. 3, pp. 1–48. de Gruyter, Berlin (1995)Google Scholar
  4. 4.
    Brady, N., Crisp, J.: Two-dimensional Artin groups with \({\rm CAT}(0)\) dimension three. In: Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), vol. 94, pp. 185–214. (2002)Google Scholar
  5. 5.
    Brady, T.: A partial order on the symmetric group and new \(K(\pi,1)\)’s for the braid groups. Adv. Math. 161(1), 20–40 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Brady, T., McCammond, J.: Braids, posets and orthoschemes. Algebraic Geom. Topol. 10(4), 2277–2314 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Brady, T., McCammond, J.P.: Three-generator Artin groups of large type are biautomatic. J. Pure Appl. Algebra 151(1), 1–9 (2000)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature, Grundlehren der Mathematischen Wissenschaften, vol. 319. Springer, Berlin (1999)CrossRefzbMATHGoogle Scholar
  9. 9.
    Brown, K.S.: Buildings. Springer, New York (1989)CrossRefzbMATHGoogle Scholar
  10. 10.
    Charney, R.: An introduction to right-angled Artin groups. Geom. Dedicata 125, 141–158 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Charney, R., Davis, M.: Singular metrics of nonpositive curvature on branched covers of Riemannian manifolds. Am. J. Math. 115(5), 929–1009 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Charney, R., Davis, M.W.: Finite \(K(\pi , 1)\)s for Artin groups. In: Prospects in Topology (Princeton, NJ, 1994), Annals of Mathematics Studies, vol. 138, pp. 110–124. Princeton University Press, Princeton (1995)Google Scholar
  13. 13.
    Grätzer, G.: Lattice Theory: Foundation. Birkhäuser/Springer Basel AG, Basel (2011)CrossRefzbMATHGoogle Scholar
  14. 14.
    Gromov, M.: Hyperbolic groups. In: Essays in Group Theory, Mathematics Science Research Institute Publicatoins, vol. 8, pp. 75–263. Springer, New York (1987)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2016

Authors and Affiliations

  1. 1.Institut Montpelliérain Alexander GrothendieckUniversité de MontpellierMontpellier Cedex 5France
  2. 2.Mathematisches InstitutUniversität BonnBonnGermany
  3. 3.Department of MathematicsKarlsruhe Institute of TechnologyKalrsruheGermany

Personalised recommendations