On Bieberbach subgroups of Bn/[Pn,Pn] and flat manifolds with cyclic holonomy [formula omitted

Abstract

Recently, Gonçalves, Guaschi and Ocampo proved that, for n≥3, Bn/[Pn,Pn] is a crystallographic group. Moreover, they showed that H˜=σ−1(H)[Pn,Pn], where σ:Bn→Sn is the natural projection, is a Bieberbach subgroup of Bn/[Pn,Pn] with holonomy group H if and only if H is a 2-subgroup of Sn. Let d≥1, in this paper we study those subgroups H˜ when H is isomorphic to Z2d. We exhibit a presentation for H˜ and a set of generators for its center. We also study in detail the holonomy representation ψ:H→Aut(Zn(n−1)2) by splitting it as a sum of irreps of H. Finally, using such decomposition of ψ, we are able to determine whether the flat manifold XH with fundamental group H˜ admits Anosov diffeomorphism and/or supports Kähler geometry.