Crystallographic groups and flat manifolds from surface braid groups

Let M be a compact surface without boundary, and n≥2. We analyse the quotient group Bn(M)/Γ2(Pn(M)) of braid Bn(M) by commutator subgroup Γ2(Pn(M)) pure Pn(M). If is different from 2-sphere S2, we prove that Bn(M)/Γ2(Pn(M))≅Pn(M)/Γ2(Pn(M))⋊φSn, crystallographic if only orientable. orientable, number results regarding structure Bn(M)/Γ2(Pn(M)). characterise finite-order elements this group, determine conjugacy classes these elements. also show there single class finite subgroups isomorphic either to Sn or certain Frobenius groups. groups whose image projection Bn(M)/Γ2(Pn(M))⟶Sn are not Bieberbach Finally, construct family G˜n,g dimension 2ng holonomy cyclic order n, Xn,g flat manifold fundamental G˜n,g, it an orientable Kähler admits Anosov diffeomorphisms.