Almost-crystallographic groups as quotients of Artin braid groups

Abstract

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 i f 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 oneto-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 ).