Almost-crystallographic groups as quotients of Artin braid groups

Autor(es)

Goncalves, Daciberg Lima ^[1] ; Guaschi, John ^[2] ; Ocampo, Oscar ^[3]

Afiliação do(s) autor(es)

^[1] Univ Sao Paulo, IME, Dept Matemat, Rua Matao 1010, BR-05508090 Sao Paulo, SP - Brazil

^[2] Normandie Univ, UNICAEN, UMR CNRS 6139, Lab Math Nicolas Oresme, CS 14032, F-14032 Caen 5 - France

^[3] Univ Fed Bahia, Dept Matemat, IME, Av Adhemar de Barros S-N, BR-40170110 Salvador, BA - Brazil

• Tipo de documento: Artigo Científico
• Fonte: Journal of Algebra; v. 524, p. 160-186, APR 15 2019.
• Citações Web of Science: 1

Resumo

Let n, k >= 3. In this paper, we analyse the quotient group B-n/Gamma(k)(P-n) of the Artin braid group B-n by the subgroup Gamma(k) (P-n) belonging to the lower central series of the Artin pure braid group P-n. We prove that it is an almost-crystallographic group. We then focus more specifically on the case k = 3. If n >= 5, and if T is an element of N is such that gcd (T, 6) = 1, we show that B-n/Gamma(3)(P-n) possesses an element of order T if and only if S-n does, and we prove that there is a one-to-one correspondence between the conjugacy classes of elements of order T in B-n/Gamma(3) (P-n) with those of elements of order Tin the symmetric group S-n. We also exhibit a presentation for the almost-crystallographic group B-n/Gamma(3)(P-n). Finally, we obtain some 4-dimensional almost-Bieberbach subgroups of B-n/Gamma(3) (P-3), we explain how to construct almost-Bieberbach subgroups of B-4/Gamma(3)(P-4) and B-3/Gamma(4)(P-3), and we exhibit explicit elements of order 5 in B-5/Gamma(3)(P-5). (C) 2019 Elsevier Inc. All rights reserved. (AU)

• Processo FAPESP: 14/50131-7 - Algebraic and topological properties of the braid groups of the real projective plane, sphere, disk, orbit configuration spaces, and relations with crystallographic groups
• Beneficiário: Daciberg Lima Gonçalves
• Modalidade de apoio: Auxílio à Pesquisa - Regular
• Processo FAPESP: 16/24707-4 - Topologia algébrica, geométrica e diferencial
• Beneficiário: Daciberg Lima Gonçalves
• Modalidade de apoio: Auxílio à Pesquisa - Temático
• Processo FAPESP: 16/50354-1 - Generalisations of configuration spaces, relations between braid and almost-crystallographic groups, and applications to the study of the Borsuk-Ulam property and multi-valued maps
• Beneficiário: Daciberg Lima Gonçalves
• Modalidade de apoio: Auxílio à Pesquisa - Regular
• Processo FAPESP: 12/24454-8 - Topologia algébrica, geométrica e diferencial
• Beneficiário: Daciberg Lima Gonçalves
• Modalidade de apoio: Auxílio à Pesquisa - Temático