On the Number of Generators of a Bieberbach Group

Abstract

In this article, we study the question of whether it is true that an n-dimensional Bieberbach group can be generated by n elements. We answer this question affirmatively in dimension less than or equal to 6, and in case the holonomy group is an elementary abelian p-group.

This result includes an alternative proof for the recent result by Yalçin, saying that whenever an elementary abelian p-group is acting freely on the n-dimensional torus, we must have that h ≤ n.

Key Words:

• Bieberbach group
• Cohomology
• Crystallographic group
• Generators
• Group extension

2000 Mathematics Subject Classification:

• Primary 20H15, 20F05
• Secondary 20E22

ACKNOWLEDGMENTS

Research supported by the Research Grant of the Research Foundation Flanders (FWO): 1.5.026.05, the Research Programmer of the Research Foundation Flanders (FWO): G0570.06, and the Research Fund of the Katholieke Universiteit Leuven.

Notes

Communicated by M. Dixon.