Volume 10, issue 4 (2010)

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Braids, posets and orthoschemes

Tom Brady and Jon McCammond

Algebraic & Geometric Topology 10 (2010) 2277–2314
Abstract

In this article 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.

Keywords
braid group, Artin group, poset, orthoscheme, nonpositive curvature
Mathematical Subject Classification 2000
Primary: 05E15, 06A06, 20F36, 20F65, 51M20
Secondary: 06A11
References
Publication
Received: 25 September 2009
Revised: 21 July 2010
Accepted: 26 August 2010
Published: 26 November 2010
Authors
Tom Brady
School of Mathematical Sciences
Dublin City University
Glasnevin, Dublin 9
Ireland
http://webpages.dcu.ie/~bradyt/
Jon McCammond
Department of Mathematics
University of California
Santa Barbara CA 93106
USA
http://www.math.ucsb.edu/~jon.mccammond