×

zbMATH — the first resource for mathematics

A quotient of the Artin braid groups related to crystallographic groups. (English) Zbl 1367.20034
Authors’ abstract: Let \(n\geq3\). In this paper, we study the quotient group \(B_n/[P_n,P_n]\) of the Artin braid group \(B_n\) by the commutator subgroup of its pure Artin braid group \(P_n\). We show that \(B_n/[P_n,P_n]\) is a crystallographic group, and in the case \(n=3\), we analyse explicitly some of its subgroups. We also prove that \(B_n/[P_n,P_n]\) possesses torsion, and we show that there is a one-to-one correspondence between the conjugacy classes of the finite-order elements of \(B_n/[P_n,P_n]\) with the conjugacy classes of the elements of odd order of the symmetric group \(S_n\), and that the isomorphism class of any abelian subgroup of odd order of \(S_n\) is realised by a subgroup of \(B_n/[P_n,P_n]\). Finally, we discuss the realisation of non-abelian subgroups of \(S_n\) of odd order as subgroups of \(B_n/[P_n,P_n]\), and we show that the Frobenius group of order 21, which is the smallest non-abelian group of odd order, embeds in \(B_n/[P_n,P_n]\) for all \(n\geq7\).

MSC:
20F36 Braid groups; Artin groups
20H15 Other geometric groups, including crystallographic groups
20F45 Engel conditions
20E07 Subgroup theorems; subgroup growth
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Berrick, A. J.; Cohen, F. R.; Wong, Y.-L.; Wu, J., Configurations, braids and homotopy groups, J. Amer. Math. Soc., 19, 265-326, (2006) · Zbl 1188.55007
[2] Brown, H.; Bülow, R.; Neubüser, J.; Wondratschek, H.; Zassenhaus, H., Crystallographic groups of four-dimensional space, Wiley Monographs in Crystallography, (1978), Wiley-Interscience · Zbl 0381.20002
[3] Charlap, L., Bieberbach groups and flat manifolds, (1986), Springer-Verlag New York · Zbl 0608.53001
[4] Conway, J. H.; Curtis, R. T.; Norton, S. P.; Parker, R. A.; Wilson, R. A., Atlas of finite groups, (1985), Oxford University Press Eynsham · Zbl 0568.20001
[5] Coxeter, H. S.M., Factor groups of the braid groups, (Proc. Fourth Canad. Math. Congress, (1957)), 95-122 · Zbl 0077.25101
[6] Dekimpe, K., Almost-Bieberbach groups: affine and polynomial structures, Lecture Notes in Mathematics, vol. 1639, (1996), Springer Berlin · Zbl 0865.20001
[7] Elrifai, E. A.; Morton, H. R., Algorithms for positive braids, Q. J. Math., 45, 479-497, (1994) · Zbl 0839.20051
[8] Gonçalves, D. L.; Guaschi, J., The braid groups of the projective plane, Algebr. Geom. Topol., 4, 757-780, (2004) · Zbl 1056.20024
[9] Hansen, V. L., Braids and coverings: selected topics, London Math. Soc. Student Text, vol. 18, (1989), Cambridge University Press · Zbl 0692.57001
[10] (Henry, N. F.M.; Lonsdale, K., International Tables for X-Ray Crystallography, vol. 1 (Symmetry Groups), (1969), International Union of Crystallography, Kynoch Press)
[11] Hoffman, M., An invariant of finite abelian groups, Amer. Math. Monthly, 94, 664-666, (1987)
[12] Johnson, D. L., Presentation of groups, London Math. Soc. Lecture Notes, vol. 22, (1976), Cambridge University Press · Zbl 0324.20040
[13] Li, J. Y.; Wu, J., Artin braid groups and homotopy groups, Proc. Lond. Math. Soc., 99, 521-556, (2009) · Zbl 1241.20042
[14] Lyndon, R., Groups and geometry, London Math. Soc. Lecture Notes, vol. 101, (1985), Cambridge University Press
[15] Marin, I., The cubic Hecke algebra on at most 5 strands, J. Pure Appl. Algebra, 216, 2754-2782, (2012) · Zbl 1266.20006
[16] Marin, I., On the representation theory of braid groups, Ann. Math. Blaise Pascal, 20, 193-260, (2013) · Zbl 1303.20047
[17] Marin, I., Crystallographic groups and flat manifolds from complex reflection groups, Geom. Dedicata, 182, 233-247, (2016) · Zbl 1353.20020
[18] Murasugi, K.; Kurpita, B. I., A study of braids, Mathematics and Its Applications, vol. 484, (1999), Kluwer Academic Publishers Dordrecht · Zbl 0938.57001
[19] Ocampo, O., Grupos de tranças brunnianas e grupos de homotopia da esfera \(\mathbb{S}^2\), (2013), Universidade de São Paulo Brazil, PhD thesis
[20] Panaite, F.; Staic, M., A quotient of the braid group related to pseudosymmetric braided categories, Pacific J. Math., 244, 155-167, (2010) · Zbl 1210.20035
[21] Tits, J., Normalisateurs de tores I : groupes de Coxeter étendus, J. Algebra, 4, 96-116, (1966) · Zbl 0145.24703
[22] Wolf, J. A., Spaces of constant curvature, vol. 372, (2011), AMS Chelsea Publishing
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.