Geometriae Dedicata

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

The 6-strand braid group is CAT(0)

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.


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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

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