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On a conjecture in Artin groups

Arye Juhász

Department of Mathematics, Technion, Israel Institute of Technology, Haifa 32000, Israel

https://doi.org/10.1142/S0218196721400105Cited by:1 (Source: Crossref)

This article is part of the issue:

Special Issue on Papers From the Conference “Semigroups and Groups, Automata, Logics” SandGAL 2019
Guest Editors: Achille Frigeri, Victoria Gould, Patrizia Longobardi and Emanuele Rodaro

Abstract

It is conjectured that an irreducible Artin group which is of infinite type has trivial center. The conjecture is known to be true for two-dimensional Artin groups and for a few other types of Artin groups. In this work, we show that the conjecture holds true for Artin groups which satisfy a condition stronger than being of infinite type. We use small cancellation theory of relative presentations.

Communicated by all the special editors

Keywords:

AMSC: 20F65, 20F35, 20F10

References

1. K. I. Appel and P. E. Schupp, Artin groups and infinite coxeter groups, Invent. Math. 72 (1983) 201–220. Crossref, Web of Science, Google Scholar
2. W. A. Bogley and S. J. Pride, Aspherical relative presentations, Proc. Edinburgh Math. Soc. 35 (1992) 1–39. Crossref, Web of Science, Google Scholar
3. E. Brieskorn and K. Saito, Artin–Gruppen and Coxeter–Gruppen, Invent. Math. 17 (1972) 245–271. Crossref, Web of Science, Google Scholar
4. R. Charney, Problems Related to Artin Groups, Problems in Geometric Group Theory, Wiki (American Institute of Mathematics) 2008. Google Scholar
5. R. Charney and M. W. Davis, The $K (π, 1)$ $K (π, 1)$ problem for hyperplane complements associated to infinite reflection groups, J. Amer. Math. 8(3) (1995) 596–627. Google Scholar
6. R. Charney and R. Morris-Wright, Artin groups of infinite type-1 trivial centers and acylindical hyperbolicity, Proc. Amer. Math. Soc. 147(9) (2019) 3675–3689. Crossref, Web of Science, Google Scholar
7. D. J. Collins and J. Huebschmann, Spherical diagrams and identities among relations, Math. Ann. 261 (1982) 155–183. Crossref, Web of Science, Google Scholar
8. P. Deligne, Les immeubles des groupes de tresses généralises, Invent. Math. 17 (1972) 272–302. Crossref, Web of Science, Google Scholar
9. M. Edjvet, Equations over groups and a theorem of Higman, Neumann and Neumann, Proc. London Math. Soc. 62 (1991) 563–589. Crossref, Web of Science, Google Scholar
10. E. Godelle, Parabolic subgroups of Artin groups of FC type, Pacific J. Math. 208(2) (2003) 243–254. Crossref, Web of Science, Google Scholar
11. E. Godelle, Artin–Tits groups with CAT(O) Deligne complex, J. Pure Appl. Algebra 208(1) (2007) 39–52. Crossref, Web of Science, Google Scholar
12. E. Godelle and L. Paris, Basic questions on Artin-Tits groups, in Configuration Spaces: Geometry, Combinatorics and Topology, Scuola Normale Superior, eds. A. Bjorner, F. Cohen, C. de Concini, C. Processi and M. Salvetti (Edizioni della Normala, Pisa, 2012), pp. 229–311. Crossref, Google Scholar
13. J. Howie, The solution of length three equations over groups, Proc. Edinburgh Math. Soc. 26(2) (1981) 165–175. Google Scholar
14. A. Juhász, On the spherical radical of Artin groups, manuscript. Google Scholar
15. A. A. Klyachko, The structure of one-relator relative presentations and their centers, J. Group Theory 12 (2009) 923–947. Crossref, Web of Science, Google Scholar
16. H. van der Lek, The homotopy type of complex hyperplane complements, Ph.D thesis, Katholieke Universiteit, Nijmegen (1983). Google Scholar
17. R. C. Lyndon, On Dehn Algorithm, Math. Ann. 106 (1966) 208–228. Crossref, Web of Science, Google Scholar
18. R. C. Lyndon and P. E. Schupp, Combinatorial Group Theory, (Springer Verlag, 1977). Crossref, Google Scholar
19. J. McCammond and R. Sulway, Artin groups of Euclidean type, Invent. Math. 210 (2017) 231–282. Crossref, Web of Science, Google Scholar