Generators of Bieberbach groups with 2-generated holonomy group
Generators of Bieberbach groups with 2-generated holonomy group
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.
Bieberbach group, Crystallographic group, Cyclic group, Generators
Chung, Ho Yiu
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Chung, Ho Yiu
b2f9e9cc-c612-453a-8c32-95ea2db9f8a4
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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Accepted/In Press date: 8 November 2018
e-pub ahead of print date: 14 November 2018
Keywords:
Bieberbach group, Crystallographic group, Cyclic group, Generators
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Local EPrints ID: 426624
URI: http://eprints.soton.ac.uk/id/eprint/426624
ISSN: 0046-5755
PURE UUID: 815bd276-9396-42f4-b7e8-c97463ff3e2a
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Date deposited: 06 Dec 2018 17:30
Last modified: 15 Mar 2024 23:15
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Author:
Ho Yiu Chung
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