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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
##### Keywords:
braid group; Artin group; poset; orthoscheme; nonpositive curvature
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