On Bieberbach subgroups of B/[P,P] and flat manifolds with cyclic holonomy Z2d

Bieberbach groups and flat manifolds with finite abelian holonomy from Artin braid groups

“…As in item (ii) let L = Γ ∩ Ker(σ). Then, taking the set of coset representatives M of L in Γ, using only the expresion given in equation ( 10) and using the Reidemeister-Schreier method we may conclude that a basis for L is the set A 1 ∪ A 2 , see equation (11) and equation (12). The proof that Γ is torsion free is in practice the same as the one given in STEP 2 of item (ii).…”

“…[P n , P n ] is a Bieberbach subgroup of B n /[P n , P n ] with holonomy group H if and only if H is a 2-subgroup of S n , where σ : B n → S n is the natural projection, see [5,Corollary 13 and Theorem 20]. In [12] was studied Bieberbach groups of the form H arising from Artin braid groups with cyclic holonomy group H = Z 2 d , the authors computed the center of the Bieberbach group H, decomposed its holonomy representation in irreps, and with this information the authors were able to determine whether the related flat manifold admits Anosov diffeomorphism and/or Kähler geometry (the last one for even dimension).…”

“…As an application, motivated by [12], in Section 4 we shall consider the case of cyclic holonomy groups of odd order, i.e. Bieberbach subgroups Γ Z q (Γ q for short) with holonomy group Z q ≤ S n of odd order.…”