# zbMATH — the first resource for mathematics

Dual Euclidean Artin groups and the failure of the lattice property. (English) Zbl 1343.20039
Summary: The irreducible Euclidean Coxeter groups that naturally act geometrically on Euclidean space are classified by the well-known extended Dynkin diagrams and these diagrams also encode the modified presentations that define the irreducible Euclidean Artin groups. These Artin groups have remained mysterious with some exceptions until very recently. Craig Squier clarified the structure of the three examples with three generators more than twenty years ago and François Digne more recently proved that two of the infinite families can be understood by constructing a dual presentation for each of these groups and showing that it forms an infinite-type Garside structure. In this article I establish that none of the remaining dual presentations for irreducible Euclidean Artin groups correspond to Garside structures because their factorization posets fail to be lattices. These are the first known examples of Artin groups where all of their dual presentations fail to form Garside structures. Nevertheless, the results presented here about the cause of this failure form the foundation for a subsequent article in which the structure of Euclidean Artin groups is finally clarified.

##### MSC:
 20F36 Braid groups; Artin groups
Full Text:
##### References:
 [1] Baumeister, Barbara; Dyer, Matthew; Stump, Christian; Wegener, Patrick, A note on the transitive Hurwitz action on decompositions of parabolic Coxeter elements · Zbl 1343.20041 [2] Bessis, David, Topology of complex reflection arrangements, available at · Zbl 1372.20036 [3] Bessis, David, The dual braid monoid, Ann. Sci. Éc. Norm. Supér. (4), 36, 5, 647-683, (2003), MR MR2032983 (2004m:20071) · Zbl 1064.20039 [4] Bridson, Martin R.; Haefliger, André, Metric spaces of non-positive curvature, Grundlehren Math. Wiss., vol. 319, (1999), Springer-Verlag Berlin, MR 2000k:53038 · Zbl 0988.53001 [5] Brady, Noel; McCammond, Jon, Factoring Euclidean isometries, Internat. J. Algebra Comput., 25, 1-2, 325-347, (2015) · Zbl 1315.51013 [6] Brady, Thomas; McCammond, Jonathan P., Three-generator Artin groups of large type are biautomatic, J. Pure Appl. Algebra, 151, 1, 1-9, (2000), MR 2001f:20076 · Zbl 1004.20023 [7] Brady, Tom; McCammond, Jon, Braids, posets and orthoschemes, Algebr. Geom. Topol., 10, 4, 2277-2314, (2010) · Zbl 1205.05246 [8] Bourbaki, Nicolas, Lie groups and Lie algebras, Elem. Math. (Berlin), (2002), Springer-Verlag Berlin, translated from the 1968 French original by Andrew Pressley, MR 1890629 (2003a:17001) · Zbl 0983.17001 [9] Crawley-Boevey, William, Exceptional sequences of representations of quivers, (Proceedings of the Sixth International Conference on Representations of Algebras, Ottawa, ON, 1992, Carleton-Ottawa Math. Lecture Note Ser., vol. 14, (1992), Carleton Univ.), 7 pp., MR 1206935 (94c:16017) · Zbl 0824.16010 [10] Charney, R.; Meier, J.; Whittlesey, K., Bestvina’s normal form complex and the homology of garside groups, Geom. Dedicata, 105, 171-188, (2004), MR MR2057250 (2005e:20057) · Zbl 1064.20044 [11] Dehornoy, Patrick; Digne, François; Godelle, Eddy; Krammer, Daan; Michel, Jean, Foundations of garside theory · Zbl 1370.20001 [12] Dehornoy, Patrick; Digne, François; Michel, Jean, Garside families and garside germs, J. Algebra, 380, 109-145, (2013), MR 3023229 · Zbl 1294.18003 [13] Digne, F., Présentations duales des groupes de tresses de type affine $$\widetilde{A}$$, Comment. Math. Helv., 81, 1, 23-47, (2006), MR 2208796 (2006k:20075) · Zbl 1143.20020 [14] Digne, F., A garside presentation for Artin-Tits groups of type $$\widetilde{C}_n$$, Ann. Inst. Fourier (Grenoble), 62, 2, 641-666, (2012), MR 2985512 · Zbl 1260.20056 [15] Dehornoy, Patrick; Paris, Luis, Gaussian groups and garside groups, two generalisations of Artin groups, Proc. Lond. Math. Soc. (3), 79, 3, 569-604, (1999), MR 2001f:20061 · Zbl 1030.20021 [16] Davey, B. A.; Priestley, H. A., Introduction to lattices and order, (2002), Cambridge University Press New York, MR MR1902334 (2003e:06001) · Zbl 1002.06001 [17] Humphreys, James E., Reflection groups and Coxeter groups, Cambridge Stud. Adv. Math., vol. 29, (1990), Cambridge University Press Cambridge, MR 1066460 (92h:20002) · Zbl 0725.20028 [18] Igusa, Kiyoshi, Exceptional sequences, braid groups and clusters, (Groups, Algebras and Applications, Contemp. Math., vol. 537, (2011), Amer. Math. Soc. Providence, RI), 227-240, MR 2799103 (2012f:16001) · Zbl 1241.16015 [19] Igusa, Kiyoshi; Schiffler, Ralf, Exceptional sequences and clusters, J. Algebra, 323, 8, 2183-2202, (2010), MR 2596373 (2011b:20118) · Zbl 1239.16019 [20] Ingalls, Colin; Thomas, Hugh, Noncrossing partitions and representations of quivers, Compos. Math., 145, 6, 1533-1562, (2009), MR 2575093 (2010m:16021) · Zbl 1182.16012 [21] Jon McCammond, Pulling apart orthogonal groups to find continuous braids, preprint, 2010. [22] McCammond, Jon, The structure of Euclidean Artin groups, in: Proceedings of the 2013 Durham Conference on Geometric and Cohomological Group Theory, in press · Zbl 1343.20039 [23] McCammond, Jon; Petersen, T. Kyle, Bounding reflection length in an affine Coxeter group, J. Algebraic Combin., 34, 4, 711-719, (2011), MR 2842917 (2012h:20089) · Zbl 1229.20034 [24] McCammond, Jon; Sulway, Robert, Artin groups of Euclidean type · Zbl 1423.20032 [25] Ringel, Claus Michael, The braid group action on the set of exceptional sequences of a hereditary Artin algebra, (Abelian Group Theory and Related Topics, Oberwolfach, 1993, Contemp. Math., vol. 171, (1994), Amer. Math. Soc. Providence, RI), 339-352, MR 1293154 (95m:16006) · Zbl 0851.16010 [26] Squier, Craig C., On certain 3-generator Artin groups, Trans. Amer. Math. Soc., 302, 1, 117-124, (1987), MR 887500 (88g:20069) · Zbl 0637.20016 [27] Snapper, Ernst; Troyer, Robert J., Metric affine geometry, Dover Books on Advanced Mathematics, (1989), Dover Publications Inc. New York, MR 1034484 (90j:51001) · Zbl 0743.51003 [28] Stanley, Richard P., Enumerative combinatorics, vol. 1, Cambridge Stud. Adv. Math., vol. 49, (1997), Cambridge University Press Cambridge, with a foreword by Gian-Carlo Rota, corrected reprint of the 1986 original, MR 98a:05001 · Zbl 0889.05001
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.