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

Abstract

Let n ≥ 3. In this paper we show that for any finite abelian subgroup G of S n the crystallographic group B n /[P n , P n ] has Bieberbach subgroups Γ G with holonomy group G. Using this approach we obtain an explicit description of the holonomy representation of the Bieberbach group Γ G . As an application, when the holonomy group is cyclic of odd order, we study the holonomy representation of Γ G and determine the existence of Anosov diffeomorphisms and Kähler geometry of the flat manifold X Γ G with fundamental group the Bieberbach group Γ G ≤ B n /[P n , P n ].

Virtual braid groups, virtual twin groups and crystallographic groups

“…torsionfree crystallographic groups) inside V B n /Γ 2 (V P n ). Follows from Proposition 2.5 that in V B n /Γ 2 (V P n ) there are infinite elements of finite order coming from B n /Γ 2 (P n ) (see [13,Theorem 3]) and also the Bieberbach groups with finite Abelian holonomy realized in B n /Γ 2 (P n ) in [13] and [21] are naturally realized in V B n /Γ 2 (V P n ).…”

“…where k θ l (r),θ l (s) = 0 if 1 ≤ θ l (r) < θ l (s) ≤ n, and k θ l (r),θ l (s) = 1 with a θ l (r),θ l (s) = a θ l (s),θ l (r) if 1 ≤ θ l (s) < θ l (r) ≤ n. Using equation (21) and following the idea of the proof of Theorem 2.9 we can conclude the proof this result.…”