# zbMATH — the first resource for mathematics

Three-dimensional FC Artin groups are CAT(0). (English) Zbl 1134.20038
Summary: Building upon earlier work of T. Brady, we construct locally CAT(0) classifying spaces for those Artin groups which are three-dimensional and which satisfy the FC (flag complex) condition. The approach is to verify the ‘link condition’ by applying gluing arguments for CAT(1) spaces and by using the curvature testing techniques of M. Elder and J. McCammond [Exp. Math. 11, No. 1, 143-158 (2002; Zbl 1042.20030)].

##### MSC:
 20F36 Braid groups; Artin groups 20F65 Geometric group theory 20F55 Reflection and Coxeter groups (group-theoretic aspects) 57M07 Topological methods in group theory
##### Keywords:
Artin groups; Coxeter groups; CAT(0) spaces
Full Text:
##### References:
 [6] Brady N., Crisp J. Two-dimensional Artin groups with CAT(0) dimension three, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), Geom. Dedicata 94. (2002), 185–214 · Zbl 1070.20043 [10] Brady T., Watts C. (2002). K($$\pi$$, 1)’s for Artin groups of finite type, Proceedings of the Conference on Geometric and Combinatorial Group Theory, Part I (Haifa, 2000), Geom. Dedicata. 94. (2002), 225–250 · Zbl 1053.20034 [18] Choi W. The existence of metrics of nonpositive curvature on the Brady–Krammer complexes for finite-type Artin groups, PhD Thesis. Texas A & M University, 2004 [20] Davis M., Moussong G. Notes on non-positively curved Polyhedra, In: Low Dimensional Topology (Eger, 1996/Budapest, 1998), Bolyai Soc. Math. Stud. 8, J’nos Bolyai Math. Soc., Budapest, 1999, pp. 11–94 · Zbl 0961.53022 [28] Hanham P. PhD Thesis, University of Southhampton, 2002 [30] van der Lek: The homotopy type of complex hyperplane arrangements, PhD thesis, University of Nijmegan, 1983 [31] Malcev A. On isomorphic matrix representations, Mat Sb. (NS) 8. (50), 405–421 · JFM 66.0088.03 [32] Moussong, G.: Hyperbolic Coxeter groups, PhD dissertation, The Ohio State University, 1988 [33] Picantin, M.: Explicit presentations for the dual Braid monoids, C.R. Acad Sci. Paris, Ser. I. (2001), 1–6 · Zbl 1020.20027
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.