×

zbMATH — the first resource for mathematics

Braids, posets and orthoschemes. (English) Zbl 1205.05246
Summary: We study the curvature properties of the order complex of a bounded graded poset under a metric that we call the “orthoscheme metric”. In addition to other results, we characterize which rank 4 posets have CAT(0) orthoscheme complexes and by applying this theorem to standard posets and complexes associated with four-generator Artin groups, we are able to show that the 5-string braid group is the fundamental group of a compact nonpositively curved space.

MSC:
05E15 Combinatorial aspects of groups and algebras (MSC2010)
06A06 Partial orders, general
20F36 Braid groups; Artin groups
20F65 Geometric group theory
51M20 Polyhedra and polytopes; regular figures, division of spaces
06A11 Algebraic aspects of posets
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] P Abramenko, K S Brown, Buildings. Theory and applications, Graduate Texts in Math. 248 (2008) · Zbl 1214.20033
[2] J Altobelli, R Charney, A geometric rational form for Artin groups of FC type, Geom. Dedicata 79 (2000) 277 · Zbl 1048.20020 · doi:10.1023/A:1005216814166
[3] D Bessis, The dual braid monoid, Ann. Sci. École Norm. Sup. \((4)\) 36 (2003) 647 · Zbl 1064.20039 · doi:10.1016/j.ansens.2003.01.001 · numdam:ASENS_2003_4_36_5_647_0 · eudml:82614
[4] A Björner, Topological methods (editors R L Graham, M Grötschel, L Lovász), Elsevier (1995) 1819 · Zbl 0851.52016
[5] B H Bowditch, Notes on locally \(\mathrm{CAT}(1)\) spaces (editors R Charney, Davis,Michael, M Shapiro), Ohio State Univ. Math. Res. Inst. Publ. 3, de Gruyter (1995) 1 · Zbl 0865.53035
[6] T Brady, Artin groups of finite type with three generators, Michigan Math. J. 47 (2000) 313 · Zbl 0996.20022 · doi:10.1307/mmj/1030132536
[7] T Brady, A partial order on the symmetric group and new \(K(\pi,1)\)’s for the braid groups, Adv. Math. 161 (2001) 20 · Zbl 1011.20040 · doi:10.1006/aima.2001.1986
[8] T Brady, J P McCammond, Three-generator Artin groups of large type are biautomatic, J. Pure Appl. Algebra 151 (2000) 1 · Zbl 1004.20023 · doi:10.1016/S0022-4049(99)00094-8
[9] T Brady, C Watt, \(K(\pi,1)\)’s for Artin groups of finite type, Geom. Dedicata 94 (2002) 225 · Zbl 1053.20034 · doi:10.1023/A:1020902610809
[10] M R Bridson, Geodesics and curvature in metric simplicial complexes (editors É Ghys, A Haefliger, A Verjovsky), World Sci. Publ. (1991) 373 · Zbl 0844.53034
[11] M R Bridson, A Haefliger, Metric spaces of non-positive curvature, Grund. der Math. Wissenschaften 319, Springer (1999) · Zbl 0988.53001
[12] W Choi, The existence of metrics of nonpositive curvature on the Brady-Krammer complexes for finite-type Artin groups, PhD thesis, Texas A&M University (2004) · proquest.umi.com
[13] H S M Coxeter, Regular complex polytopes, Cambridge Univ. Press (1991) · Zbl 0732.51002
[14] M Elder, J McCammond, Curvature testing in \(3\)-dimensional metric polyhedral complexes, Experiment. Math. 11 (2002) 143 · Zbl 1042.20030 · doi:10.1080/10586458.2002.10504476 · eudml:50846
[15] M Elder, J McCammond, J Meier, Combinatorial conditions that imply word-hyperbolicity for \(3\)-manifolds, Topology 42 (2003) 1241 · Zbl 1029.57013 · doi:10.1016/S0040-9383(02)00100-3 · arxiv:math/0301057
[16] A Lytchak, Rigidity of spherical buildings and joins, Geom. Funct. Anal. 15 (2005) 720 · Zbl 1083.53044 · doi:10.1007/s00039-005-0519-6
[17] J McCammond, Noncrossing partitions in surprising locations, Amer. Math. Monthly 113 (2006) 598 · Zbl 1179.05015 · doi:10.2307/27642003
[18] R P Stanley, Enumerative combinatorics. Vol. 1, Cambridge Stud. in Adv. Math. 49, Cambridge Univ. Press (1997) · Zbl 0889.05001
[19] G M Ziegler, Lectures on polytopes, Graduate Texts in Math. 152, Springer (1995) · Zbl 0823.52002
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.