Problems on Bieberbach groups and flat manifolds

We present about twenty conjectures, problems and questions about flat manifolds. Many of them build the bridges between the flat world and representation theory of the finite groups, hyperbolic geometry and dynamical systems. We shall present here some conjectures, problems and open questions related to flat manifolds. By flat manifold we understand a compact closed (without boundary) Riemannian manifold with sectional curvature equal to zero. It is well known (see for example [35]) that any flat manifold M can be considered as an orbit space R/Γ, where Γ is a discrete, torsion free and cocompact subgroup of the group E(n) = O(n)⋉R = IsomR. It is easy to see that Γ = π1(M ). If we remove the assumption that Γ is torsion free we obtain a crystallographic group. Hence we also consider the problems about them. From the Bieberbach theorems (see [4]) any crystallographic group has a maximal free abelian subgroup (subgroup of all translations) of finite index. Hence for any such group Γ ⊂ E(n) we have a short exact sequence 0 → Z → Γ → H → 0. Here H is a finite group which in case of torsion free crystallographic group (Bieberbach group) is isomorphic to the holonomy group of manifold R/Γ. We call it the holonomy group of Γ. Since the subgroup Z is maximal abelian,