Generators of Bieberbach groups with 2-generated holonomy group

TL;DR

This paper proves that n-dimensional Bieberbach groups with certain 2-generated or specific holonomy groups can be generated by n elements, confirming a conjecture in these cases.

Contribution

It establishes the conjecture for groups with 2-generated holonomy or holonomy of order not divisible by 2 or 3, and shows cyclic holonomy groups of larger order are generated by n-1 elements.

Findings

Bieberbach groups with 2-generated holonomy can be generated by n elements.

Groups with cyclic holonomy of order > 2 are generated by n-1 elements.

The conjecture holds for holonomy groups with order not divisible by 2 or 3.

Abstract

An n-dimensional Bieberbach group is the fundamental group of a closed flat $n$-dimensional manifold. K. Dekimpe and P. Penninckx conjectured that an n-dimensional Bieberbach group can be generated by n elements. In this paper, we show that the conjecture is true if the holonomy group is 2-generated (e.g. dihedral group, quaternion group or simple group) or the order of holonomy group is not divisible by 2 or 3. In order to prove this, we show that an n-dimensional Bieberbach group with cyclic holonomy group of order larger than two can be generated by n-1 elements.