2018/05/29 by Gonçalves, Daciberg Lima, Guaschi, John, Ocampo, Oscar
#FOS: Mathematics #Geometric Topology (math.GT) #Group Theory (math.GR)
paper · doi:10.48550/arxiv.1805.11376
Let n, k ≥ 3. In this paper, we analyse the quotient group B_n/Γ_k(P_n) of the Artin braid group B_n by the subgroup Γ_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 τ∈ N is such that gcd(τ, 6) = 1, we show that B_n/Γ_3 (P_n) possesses torsion τ 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 τ in B_n/Γ_3 (P_n) with those of elements of order τ in the symmetric group S_n. We also exhibit a presentation for the almost-crystallographic group B_n/Γ_3 (P_n). Finally, we obtain some 4-dimensional almost-Bieberbach subgroups of B_3/Γ_3 (P_3), we explain how to obtain almost-Bieberbach subgroups of B_4/Γ_3(P_4) and B_3/Γ_4(P_3), and we exhibit explicit elements of order 5 in B_5/Γ_3 (P_5).