zbMATH — the first resource for mathematics

The 6-strand braid group is \(\mathrm{CAT}(0)\). (English) Zbl 1347.20044
Summary: We show that braid groups with at most 6 strands are \(\mathrm{CAT}(0)\) 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 \(\mathrm{CAT}(0)\), giving a partial answer to a conjecture of Brady and McCammond.

20F65 Geometric group theory
20F36 Braid groups; Artin groups
20E42 Groups with a \(BN\)-pair; buildings
57M07 Topological methods in group theory
Full Text: DOI arXiv
[1] Abramenko, P., Brown, K.S.: Buildings. Theory and Applications, Graduate Texts in Mathematics, vol. 248. Springer, New York (2008) · Zbl 1214.20033
[2] Bell, RW, Three-dimensional FC Artin groups are CAT(0), Geom. Dedicata, 113, 21-53, (2005) · Zbl 1134.20038
[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) · Zbl 0865.53035
[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) · Zbl 1070.20043
[5] Brady, T, A partial order on the symmetric group and new \(K(π,1)\)’s for the braid groups, Adv. Math., 161, 20-40, (2001) · Zbl 1011.20040
[6] Brady, T; McCammond, J, Braids, posets and orthoschemes, Algebraic Geom. Topol., 10, 2277-2314, (2010) · Zbl 1205.05246
[7] Brady, T; McCammond, JP, Three-generator Artin groups of large type are biautomatic, J. Pure Appl. Algebra, 151, 1-9, (2000) · Zbl 1004.20023
[8] Bridson, M.R., Haefliger, A.: Metric Spaces of Non-positive Curvature, Grundlehren der Mathematischen Wissenschaften, vol. 319. Springer, Berlin (1999) · Zbl 0988.53001
[9] Brown, K.S.: Buildings. Springer, New York (1989) · Zbl 0715.20017
[10] Charney, R, An introduction to right-angled Artin groups, Geom. Dedicata, 125, 141-158, (2007) · Zbl 1152.20031
[11] Charney, R; Davis, M, Singular metrics of nonpositive curvature on branched covers of Riemannian manifolds, Am. J. Math., 115, 929-1009, (1993) · Zbl 0804.53056
[12] Charney, R., Davis, M.W.: Finite \(K(π , 1)\)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) · Zbl 0930.55006
[13] Grätzer, G.: Lattice Theory: Foundation. Birkhäuser/Springer Basel AG, Basel (2011) · Zbl 1233.06001
[14] Gromov, M.: Hyperbolic groups. In: Essays in Group Theory, Mathematics Science Research Institute Publicatoins, vol. 8, pp. 75-263. Springer, New York (1987)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.