The mysterious geometry of Artin groups

McCammond, Jon ¹

¹ Dept. of Math., University of California, Santa Barbara, CA 93106

Winter Braids VII (Caen, 2017), Winter Braids Lecture Notes (2017), Exposé no. 1, 30 p.

Résumé

Artin groups are easily defined but most of them are poorly understood. In this survey I try to highlight precisely where the problems begin. The first part reviews the close connection between Coxeter groups and Artin groups as well as the associated topological spaces used to investigate them. The second part describes the location of the border between the Artin groups we understand at a very basic level and those that remain fundamentally mysterious. The third part highlights those collections of Artin groups (and their relatives) that are not currently understood but which we are likely to understand sometime soon.

DOI : 10.5802/wbln.17

Classification : 20F36, 20F55
Mots clés : Artin groups

Affiliations des auteurs :

McCammond, Jon ¹

¹ Dept. of Math., University of California, Santa Barbara, CA 93106

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McCammond, Jon. The mysterious geometry of Artin groups, dans Winter Braids VII (Caen, 2017), Winter Braids Lecture Notes (2017), Exposé no. 1, 30 p. doi : 10.5802/wbln.17. http://archive.numdam.org/articles/10.5802/wbln.17/

Bibliographie
Cité par

[1] Agol, Ian Criteria for virtual fibering, J. Topol., Volume 1 (2008) no. 2, pp. 269-284 | DOI | MR

[2] Agol, Ian The virtual Haken conjecture, Doc. Math., Volume 18 (2013), pp. 1045-1087 (With an appendix by Agol, Daniel Groves, and Jason Manning) | MR

[3] Allison, Bruce N.; Azam, Saeid; Berman, Stephen; Gao, Yun; Pianzola, Arturo Extended affine Lie algebras and their root systems, Mem. Amer. Math. Soc., Volume 126 (1997) no. 603, x+122 pages | DOI | MR

[4] Appel, K. I.; Schupp, P. E. and Artin groups and infinite Coxeter groups, Invent. Math., Volume 72 (1983) no. 2, pp. 201-220 | DOI | MR

[5] Bestvina, Mladen; Brady, Noel Morse theory and finiteness properties of groups, Invent. Math., Volume 129 (1997) no. 3, pp. 445-470 | DOI | MR

[6] Bigelow, Stephen J. Braid groups are linear, J. Amer. Math. Soc., Volume 14 (2001) no. 2, pp. 471-486 | DOI | MR

[7] Björner, Anders; Brenti, Francesco Combinatorics of Coxeter groups, Graduate Texts in Mathematics, 231, Springer, New York, 2005, xiv+363 pages | MR

[8] Blair, Ryan; Ottman, Ryan A decomposition theorem for higher rank Coxeter groups, Comm. Algebra, Volume 41 (2013) no. 7, pp. 2508-2518 | DOI | MR

[9] Bridson, Martin R.; Haefliger, André Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 319, Springer-Verlag, Berlin, 1999, xxii+643 pages | MR

[10] Brieskorn, Egbert; Saito, Kyoji Artin-Gruppen und Coxeter-Gruppen, Invent. Math., Volume 17 (1972), pp. 245-271 | MR

[11] Callegaro, Filippo; Moroni, Davide; Salvetti, Mario Cohomology of affine Artin groups and applications, Trans. Amer. Math. Soc., Volume 360 (2008) no. 8, pp. 4169-4188 | DOI | MR

[12] Callegaro, Filippo; Moroni, Davide; Salvetti, Mario Cohomology of Artin groups of type ${\tilde{A}}_{n}, {\tilde{B}}_{n}$ and applications, Groups, homotopy and configuration spaces (Geom. Topol. Monogr.), Volume 13, Geom. Topol. Publ., Coventry, 2008, pp. 85-104 | DOI | MR

[13] Callegaro, Filippo; Moroni, Davide; Salvetti, Mario The $K (π, 1)$ problem for the affine Artin group of type ${\tilde{B}}_{n}$ and its cohomology, J. Eur. Math. Soc. (JEMS), Volume 12 (2010) no. 1, pp. 1-22 | DOI | MR

[14] Charney, Ruth The Deligne complex for the four-strand braid group, Trans. Amer. Math. Soc., Volume 356 (2004) no. 10, pp. 3881-3897 | DOI | MR

[15] Charney, Ruth; Paris, Luis Convexity of parabolic subgroups in Artin groups, Bull. Lond. Math. Soc., Volume 46 (2014) no. 6, pp. 1248-1255 | DOI | MR

[16] Chermak, Andrew Locally non-spherical Artin groups, J. Algebra, Volume 200 (1998) no. 1, pp. 56-98 | DOI | MR

[17] Cohen, Arjeh M. Finite complex reflection groups, Ann. Sci. École Norm. Sup. (4), Volume 9 (1976) no. 3, pp. 379-436 | MR

[18] Cohen, Arjeh M.; Wales, David B. Linearity of Artin groups of finite type, Israel J. Math., Volume 131 (2002), pp. 101-123 | DOI | MR

[19] Coté, Ben; McCammond, Jon A complex euclidean reflection group with an elegant complement complex (arXiv:1707.06624 [math.GR])

[20] Crisp, John Injective maps between Artin groups, Geometric group theory down under (Canberra, 1996), de Gruyter, Berlin, 1999, pp. 119-137 | MR

[21] Crisp, John; Paris, Luis The solution to a conjecture of Tits on the subgroup generated by the squares of the generators of an Artin group, Invent. Math., Volume 145 (2001) no. 1, pp. 19-36 | DOI | MR

[22] Davis, Michael W. Groups generated by reflections and aspherical manifolds not covered by Euclidean space, Ann. of Math. (2), Volume 117 (1983) no. 2, pp. 293-324 | DOI | MR

[23] Davis, Michael W. The geometry and topology of Coxeter groups, London Mathematical Society Monographs Series, 32, Princeton University Press, Princeton, NJ, 2008, xvi+584 pages | MR

[24] Deligne, Pierre Les immeubles des groupes de tresses généralisés, Invent. Math., Volume 17 (1972), pp. 273-302 | MR

[25] Digne, François On the linearity of Artin braid groups, J. Algebra, Volume 268 (2003) no. 1, pp. 39-57 | DOI | MR

[26] Godelle, Eddy; Paris, Luis Basic questions on Artin-Tits groups, Configuration spaces (CRM Series), Volume 14, Ed. Norm., Pisa, 2012, pp. 299-311 | DOI | MR

[27] Goryunov, Victor; Man, Show Han The complex crystallographic groups and symmetries of $J_{10}$ , Singularity theory and its applications (Adv. Stud. Pure Math.), Volume 43, Math. Soc. Japan, Tokyo, 2006, pp. 55-72 | MR

[28] Huang, Jingyin; Jankiewicz, Kasia; Przytycki, Piotr Cocompactly cubulated 2-dimensional Artin groups, Comment. Math. Helv., Volume 91 (2016) no. 3, pp. 519-542 | DOI | MR

[29] Humphreys, James E. Reflection groups and Coxeter groups, Cambridge Studies in Advanced Mathematics, 29, Cambridge University Press, Cambridge, 1990, xii+204 pages | MR

[30] Krammer, Daan The braid group $B_{4}$ is linear, Invent. Math., Volume 142 (2000) no. 3, pp. 451-486 | DOI | MR

[31] Krammer, Daan Braid groups are linear, Ann. of Math. (2), Volume 155 (2002) no. 1, pp. 131-156 | DOI | MR

[32] Lehrer, Gustav I.; Taylor, Donald E. Unitary reflection groups, Australian Mathematical Society Lecture Series, 20, Cambridge University Press, Cambridge, 2009, viii+294 pages | MR

[33] McCammond, Jon Combinatorial descriptions of multi-vertex 2-complexes, Illinois J. Math., Volume 54 (2010) no. 1, pp. 137-154 http://projecteuclid.org/euclid.ijm/1299679742 | MR

[34] McCammond, Jon The structure of Euclidean Artin groups, London Mathematical Society Lecture Note Series, Cambridge University Press (2017), 82Ð114 pages | DOI

[35] McCammond, Jon; Sulway, Robert Artin groups of Euclidean type, Inventiones mathematicae, Volume 210 (2017) no. 1, pp. 231-282 | DOI

[36] Paris, Luis Artin monoids inject in their groups, Comment. Math. Helv., Volume 77 (2002) no. 3, pp. 609-637 | MR

[37] Paris, Luis $K (π, 1)$ conjecture for Artin groups, Ann. Fac. Sci. Toulouse Math. (6), Volume 23 (2014) no. 2, pp. 361-415 | DOI | MR

[38] Paris, Luis Lectures on Artin groups and the $K (π, 1)$ conjecture, Groups of exceptional type, Coxeter groups and related geometries (Springer Proc. Math. Stat.), Volume 82, Springer, New Delhi, 2014, pp. 239-257 | DOI | MR

[39] Popov, V. L. Discrete complex reflection groups, Communications of the Mathematical Institute, Rijksuniversiteit Utrecht, 15, Rijksuniversiteit Utrecht, Mathematical Institute, Utrecht, 1982, 89 pages | MR

[40] Saito, Kyoji Einfach-elliptische Singularitäten, Invent. Math., Volume 23 (1974), pp. 289-325 | DOI | MR

[41] Saito, Kyoji Extended affine root systems. I. Coxeter transformations, Publ. Res. Inst. Math. Sci., Volume 21 (1985) no. 1, pp. 75-179 | DOI | MR

[42] Saito, Kyoji Extended affine root systems. II. Flat invariants, Publ. Res. Inst. Math. Sci., Volume 26 (1990) no. 1, pp. 15-78 | DOI | MR

[43] Saito, Kyoji Extended affine root systems. V. Elliptic eta-products and their Dirichlet series, Proceedings on Moonshine and related topics (Montréal, QC, 1999) (CRM Proc. Lecture Notes), Volume 30, Amer. Math. Soc., Providence, RI (2001), pp. 185-222 | MR

[44] Saito, Kyoji; Takebayashi, Tadayoshi Extended affine root systems. III. Elliptic Weyl groups, Publ. Res. Inst. Math. Sci., Volume 33 (1997) no. 2, pp. 301-329 | DOI | MR

[45] Saito, Kyoji; Yoshii, Daigo Extended affine root system. IV. Simply-laced elliptic Lie algebras, Publ. Res. Inst. Math. Sci., Volume 36 (2000) no. 3, pp. 385-421 | DOI | MR

[46] Shephard, G. C.; Todd, J. A. Finite unitary reflection groups, Canadian J. Math., Volume 6 (1954), pp. 274-304 | MR

[47] Tits, Jacques Œuvres/Collected works. Vol. I, II, III, IV, Heritage of European Mathematics, European Mathematical Society (EMS), Zürich, 2013, Vol.I: xcviii+879 pp.; II: xii+952 pp.; III: xii+986 pp.; IV: xii+1020 pages (Edited by Francis Buekenhout, Bernhard Matthias Mühlherr, Jean-Pierre Tignol and Hendrik Van Maldeghem) | MR

[48] van der Lek, Harm The homotopy type of complex hyperplane complements, Ph. D. Thesis, University of Nijmegen (1983)

[49] Wise, Daniel T. From riches to raags: 3-manifolds, right-angled Artin groups, and cubical geometry, CBMS Regional Conference Series in Mathematics, 117, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 2012, xiv+141 pages | DOI | MR

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