A quotient of the Artin braid groups related to crystallographic groups

Daciberg Lima Gonçalves (1) , John Guaschi (2, 3, 4) , Oscar Ocampo (5)

1 USP - Universidade de São Paulo = University of São Paulo
2 LMNO - Laboratoire de Mathématiques Nicolas Oresme
3 UNICAEN - Université de Caen Normandie
4 NU - Normandie Université
5 UFBA - Universidade Federal da Bahia

Let n be greater than or equal to 3. 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 greater than or equal to 7.

Domaines: Théorie des groupes [math.GR]; Topologie géométrique [math.GT]

Daciberg Lima Gonçalves, John Guaschi, Oscar Ocampo. A quotient of the Artin braid groups related to crystallographic groups. Journal of Algebra, 2017, 474, pp.393-423. ⟨10.1016/j.jalgebra.2016.11.003⟩. ⟨hal-01131589⟩