Artin groups of infinite type: trivial centers and acylindrical hyperbolicity. (English) Zbl 1483.20068
Summary: While finite type Artin groups and right-angled Artin groups are well understood, little is known about more general Artin groups. In this paper, we use the action of an infinite type Artin group \(A_{\Gamma}\) on a CAT(0) cube complex to prove that \(A_{\Gamma}\) has trivial center providing \(\Gamma\) is not the star of a single vertex, and is acylindrically hyperbolic providing \(\Gamma\) is not a join.
MSC:
| 20F36 | Braid groups; Artin groups |
| 20F65 | Geometric group theory |
| 20F67 | Hyperbolic groups and nonpositively curved groups |
| 57M07 | Topological methods in group theory |
References:
| [1] | Altobelli, Joe; Charney, Ruth, A geometric rational form for Artin groups of FC type, Geom. Dedicata, 79, 3, 277-289 (2000) · Zbl 1048.20020 · doi:10.1023/A:1005216814166 |
| [2] | Behrstock, Jason; Dru\c{t}u, Cornelia; Mosher, Lee, Thick metric spaces, relative hyperbolicity, and quasi-isometric rigidity, Math. Ann., 344, 3, 543-595 (2009) · Zbl 1220.20037 · doi:10.1007/s00208-008-0317-1 |
| [3] | Bestvina, Mladen; Bromberg, Ken; Fujiwara, Koji, Constructing group actions on quasi-trees and applications to mapping class groups, Publ. Math. Inst. Hautes \'{E}tudes Sci., 122, 1-64 (2015) · Zbl 1372.20029 · doi:10.1007/s10240-014-0067-4 |
| [4] | Brady, Tom; McCammond, Jon, Braids, posets and orthoschemes, Algebr. Geom. Topol., 10, 4, 2277-2314 (2010) · Zbl 1205.05246 · doi:10.2140/agt.2010.10.2277 |
| [5] | Bridson, Martin R.; Haefliger, Andr\'{e}, Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 319, xxii+643 pp. (1999), Springer-Verlag, Berlin · Zbl 0988.53001 · doi:10.1007/978-3-662-12494-9 |
| [6] | Calvez, Matthieu; Wiest, Bert, Acylindrical hyperbolicity and Artin-Tits groups of spherical type, Geom. Dedicata, 191, 199-215 (2017) · Zbl 1423.20028 · doi:10.1007/s10711-017-0252-y |
| [7] | Charney, Ruth; Davis, Michael W., Finite \(K(\pi, 1)\) s for Artin groups. Prospects in topology, Princeton, NJ, 1994, Ann. of Math. Stud. 138, 110-124 (1995), Princeton Univ. Press, Princeton, NJ · Zbl 0930.55006 |
| [8] | Charney, Ruth; Davis, Michael W., The \(K(\pi,1)\)-problem for hyperplane complements associated to infinite reflection groups, J. Amer. Math. Soc., 8, 3, 597-627 (1995) · Zbl 0833.51006 · doi:10.2307/2152924 |
| [9] | Ch-Ma I. Chatterji and A. Martin, A note on the acylindrical hyperbolicity of groups acting on CAT(0) cube complexes, arXiv:1610.06864 (2016). |
| [10] | Dehornoy, Patrick, Multifraction reduction I: The 3-Ore case and Artin-Tits groups of type FC, J. Comb. Algebra, 1, 2, 185-228 (2017) · Zbl 1422.20020 · doi:10.4171/JCA/1-2-3 |
| [11] | Dehornoy, Patrick, Multifraction reduction II: conjectures for Artin-Tits groups, J. Comb. Algebra, 1, 3, 229-287 (2017) · Zbl 1422.20021 · doi:10.4171/JCA/1-3-1 |
| [12] | Deligne, Pierre, Les immeubles des groupes de tresses g\'{e}n\'{e}ralis\'{e}s, Invent. Math., 17, 273-302 (1972) · Zbl 0238.20034 · doi:10.1007/BF01406236 |
| [13] | Ellis, Graham; Sk\"{o}ldberg, Emil, The \(K(\pi,1)\) conjecture for a class of Artin groups, Comment. Math. Helv., 85, 2, 409-415 (2010) · Zbl 1192.55011 · doi:10.4171/CMH/200 |
| [14] | Gen2 A. Genevois, Acylindrical action on the hyperplanes of a CAT(0) cube complex, arXiv:1610.08759 (2016). |
| [15] | Gen A. Genevois, Hyperbolicities in CAT(0) cube complexes, arXiv:1709.08843 (2017). · Zbl 1472.20090 |
| [16] | Godelle, Eddy, Parabolic subgroups of Artin groups of type FC, Pacific J. Math., 208, 2, 243-254 (2003) · Zbl 1063.20040 · doi:10.2140/pjm.2003.208.243 |
| [17] | Godelle, Eddy; Paris, Luis, Basic questions on Artin-Tits groups. Configuration spaces, CRM Series 14, 299-311 (2012), Ed. Norm., Pisa · Zbl 1282.20036 · doi:10.1007/978-88-7642-431-1\_13 |
| [18] | Godelle, Eddy; Paris, Luis, \(K(\pi,1)\) and word problems for infinite type Artin-Tits groups, and applications to virtual braid groups, Math. Z., 272, 3-4, 1339-1364 (2012) · Zbl 1300.20045 · doi:10.1007/s00209-012-0989-9 |
| [19] | Hae Thomas Haettel, Virtually cocompactly cubulated Artin-Tits groups, arXiv:1509.08711 (2015). · Zbl 1499.20103 |
| [20] | Haettel, Thomas; Kielak, Dawid; Schwer, Petra, The 6-strand braid group is \({\rm CAT}(0)\), Geom. Dedicata, 182, 263-286 (2016) · Zbl 1347.20044 · doi:10.1007/s10711-015-0138-9 |
| [21] | Juh Arye Juh\'asz, On the spherical radical of Artin groups, preprint, 2018. |
| [22] | Martin, Alexandre, On the acylindrical hyperbolicity of the tame automorphism group of \({\rm SL}_2(\mathbb{C})\), Bull. Lond. Math. Soc., 49, 5, 881-894 (2017) · Zbl 1439.20046 · doi:10.1112/blms.12071 |
| [23] | Osin, D., Acylindrically hyperbolic groups, Trans. Amer. Math. Soc., 368, 2, 851-888 (2016) · Zbl 1380.20048 · doi:10.1090/tran/6343 |
| [24] | Paris, Luis, Parabolic subgroups of Artin groups, J. Algebra, 196, 2, 369-399 (1997) · Zbl 0926.20022 · doi:10.1006/jabr.1997.7098 |
| [25] | Paris, Luis, \(K(\pi,1)\) conjecture for Artin groups, Ann. Fac. Sci. Toulouse Math. (6), 23, 2, 361-415 (2014) · Zbl 1306.20040 · doi:10.5802/afst.1411 |
| [26] | Tits, J., Normalisateurs de tores. I. Groupes de Coxeter \'{e}tendus, J. Algebra, 4, 96-116 (1966) · Zbl 0145.24703 · doi:10.1016/0021-8693(66)90053-6 |
| [27] | vdL H. van der Lek, The homotopy type of complex hyperplane complements, Ph.D. Thesis, University of Nijmegen, Nijmegen, Netherlands, 1983. |
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