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ACYLINDRICAL HYPERBOLICITY OF ARTIN–TITS GROUPS ASSOCIATED WITH TRIANGLE-FREE GRAPHS AND CONES OVER SQUARE-FREE BIPARTITE GRAPHS

Part of: Special aspects of infinite or finite groups

Published online by Cambridge University Press:  01 December 2020

MOTOKO KATO  and
SHIN-ICHI OGUNI
MOTOKO KATO
Affiliation:
Graduate School of Science and Engineering, Mathematics, Physics and Earth Sciences, Ehime University, 2-5 Bunkyo-cho, Matsuyama, Ehime 790-8577Japan, e-mails: kato.motoko.yy@ehime-u.ac.jp, oguni.shinichi.mb@ehime-u.ac.jp
SHIN-ICHI OGUNI
Affiliation:
Graduate School of Science and Engineering, Mathematics, Physics and Earth Sciences, Ehime University, 2-5 Bunkyo-cho, Matsuyama, Ehime 790-8577Japan, e-mails: kato.motoko.yy@ehime-u.ac.jp, oguni.shinichi.mb@ehime-u.ac.jp
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Abstract

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It is conjectured that the central quotient of any irreducible Artin–Tits group is either virtually cyclic or acylindrically hyperbolic. We prove this conjecture for Artin–Tits groups that are known to be CAT(0) groups by a result of Brady and McCammond, that is, Artin–Tits groups associated with graphs having no 3-cycles and Artin–Tits groups of almost large type associated with graphs admitting appropriate directions. In particular, the latter family contains Artin–Tits groups of large type associated with cones over square-free bipartite graphs.

MSC classification

Primary: 20F65: Geometric group theory
Secondary: 20F36: Braid groups; Artin groups 20F67: Hyperbolic groups and nonpositively curved groups
Type
Research Article
Information
Glasgow Mathematical Journal , Volume 64 , Issue 1 , January 2022 , pp. 51 - 64
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s), 2020. Published by Cambridge University Press on behalf of Glasgow Mathematical Journal Trust

Footnotes

†

The first author is supported by JSPS KAKENHI Grant-in-Aid for Research Activity Start-up, Grant Number 19K23406 and JSPS KAKENHI Grant-in-Aid for Young Scientists, Grant Number 20K14311.

‡

The second author is supported by JSPS KAKENHI Grant-in-Aid for Young Scientists (B), Grant Number 16K17595 and 20K03590.

References

Ballmann, W., Singular spaces of nonpositive curvature, in Sur les groupes hyperboliques d’après Mikhael Gromov (Bern, 1988), Progress in Mathematics, vol. 83 (Birkhäuser Boston, Boston, MA, 1990), 189–201.10.1007/978-1-4684-9167-8_10CrossRefGoogle Scholar
Bestvina, M., Bromberg, K. and Fujiwara, K., Constructing group actions on quasi-trees and applications to mapping class groups, Publ. Math. Inst. Hautes Études Sci. 122 (2015), 1–64.10.1007/s10240-014-0067-4CrossRefGoogle Scholar
Bestvina, M. and Fujiwara, K., Bounded cohomology of subgroups of mapping class groups, Geom. Topol. 6 (2002), 69–89.10.2140/gt.2002.6.69CrossRefGoogle Scholar
Bowditch, B. H., Tight geodesics in the curve complex, Invent. Math. 171(2) (2008), 281–300.10.1007/s00222-007-0081-yCrossRefGoogle Scholar
Brady, T. and McCammond, J. P., Three-generator Artin groups of large type are biautomatic, J. Pure Appl. Algebra 151(1) (2000), 1–9.10.1016/S0022-4049(99)00094-8CrossRefGoogle Scholar
Bridson, M. R. and Haefliger, A., Metric spaces of non-positive curvature, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 319 (Springer-Verlag, Berlin, 1999).Google Scholar
Brieskorn, E. and Saito, K., Artin-Gruppen und Coxeter-Gruppen, Invent. Math. 17 (1972), 245–271.10.1007/BF01406235CrossRefGoogle Scholar
Calvez, M. and Wiest, B., Acylindrical hyperbolicity and Artin-Tits groups of spherical type, Geom. Dedicata 191 (2017), 199–215.10.1007/s10711-017-0252-yCrossRefGoogle Scholar
Caprace, P.-E. and Sageev, M., Rank rigidity for CAT(0) cube complexes, Geom. Funct. Anal. 21(4) (2011), 851–891.Google Scholar
Charney, R., Problems related to Artin groups (2008). http://people.brandeis.edu/~charney/papers/Artin_probs.pdf.Google Scholar
Charney, R. and Morris-Wright, R., Artin groups of infinite type: trivial centers and acylindrical hyperbolicity, Proc. Amer. Math. Soc. 147(9) (2019), 3675–3689.10.1090/proc/14503CrossRefGoogle Scholar
Chatterji, I. and Martin, A., A note on the acylindrical hyperbolicity of groups acting on CAT(0) cube complexes, in Beyond hyperbolicity (Hagen, M., Webb, R. and Wilton, H., Editors), London Mathematical Society Lecture Note Series, vol. 454 (Cambridge University Press, Cambridge, 2019), 160–178.Google Scholar
Dahmani, F., Guirardel, V. and Osin, D., Hyperbolically embedded subgroups and rotating families in groups acting on hyperbolic spaces, Mem. Amer. Math. Soc. 245(1156) (2017), v+152.Google Scholar
Deligne, P., Les immeubles des groupes de tresses généralisés, Invent. Math. 17 (1972), 273–302.10.1007/BF01406236CrossRefGoogle Scholar
Godelle, E. and Paris, L., Basic questions on Artin-Tits groups, in Configuration spaces, CRM Series, vol. 14 (Ed. Norm., Pisa, 2012), 299–311.Google Scholar
Haettel, T., XXL type artin groups are CAT(0) and acylindrically hyperbolic, preprint, 2019, arXiv:1905.11032.Google Scholar
Hamenstädt, U., Bounded cohomology and isometry groups of hyperbolic spaces, J. Eur. Math. Soc. (JEMS) 10(2) (2008), 315–349.10.4171/JEMS/112CrossRefGoogle Scholar
Kim, S.-H. and Koberda, T., The geometry of the curve graph of a right-angled Artin group, Int. J. Algebra Comput. 24(2) (2014), 121–169.10.1142/S021819671450009XCrossRefGoogle Scholar
Martin, A. and Przytycki, P., Acylindrical actions for two-dimensional Artin groups of hyperbolic type, preprint, 2019, arXiv:1906.03154.Google Scholar
Moussong, G., Hyperbolic Coxeter groups. Thesis (PhD)–The Ohio State University (ProQuest LLC, Ann Arbor, MI, 1988).Google Scholar
Osin, D., Acylindrically hyperbolic groups, Trans. Amer. Math. Soc. 368(2) (2016), 851–888.10.1090/tran/6343CrossRefGoogle Scholar
Sisto, A., Contracting elements and random walks, J. Reine Angew. Math. 742 (2018), 79–114.10.1515/crelle-2015-0093CrossRefGoogle Scholar