On the Number of Generators of a Bieberbach Group

Generators of Bieberbach groups with 2-generated holonomy group

“…By [1,8] and the three main theorems in this paper, the Conjecture 1.1 is still open for certain cases of holonomy group where the minimal number of generators has at least three elements. For example, the case where the holonomy group is a 2-group or the order of holonomy group is even.…”

“…In this paper, we focus on the below conjecture. Conjecture 1.1 [8, Let be an n-dimensional Bieberbach group. Then the minimum number of generators of is less than or equal to n.…”

“…The conjecture was solved for some special cases. For example, the conjecture is true if the holonomy group is an odd prime p-group (see [1]), or the holonomy group is an elementary abelian p-group (see [8]). On the other hand, by a computer program namely CARAT, it has been checked that the conjecture is true if the Bieberbach group has dimension less than 7 (see [5]).…”

“…Hence if π 1 (M) can be generated by n elements, then the minimal number of generators of G should not be larger than n. For instance, we know that (Z/2Z) n+1 cannot act freely on T n for n ≥ 1. (see [8,10]). Let G be a group and M be a ZG-module.…”

On generators of crystallographic groups and actions on flat orbifolds

“…For flat manifolds the existing bounds are considerably lower. In [4], Dekimpe and Penninckx proved that every n-dimensional crystallographic group with elementary abelian p-group holonomy, for any odd prime p, is generated by n elements. They also showed that the same bound holds for every n-dimensional Bieberbach group with elementary abelian 2-group holonomy.…”