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arXiv · 1807.07446

Generators of Bieberbach groups with 2-generated holonomy group

Ho Yiu Chung
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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.

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Ho Yiu Chung. 2018-07-19. Generators of Bieberbach groups with 2-generated holonomy group. https://arxiv.org/abs/1807.07446

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