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Generators of Bieberbach groups with 2-generated holonomy group

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
0046-5755
Chung, Ho Yiu
b2f9e9cc-c612-453a-8c32-95ea2db9f8a4
Chung, Ho Yiu
b2f9e9cc-c612-453a-8c32-95ea2db9f8a4

Chung, Ho Yiu (2018) Generators of Bieberbach groups with 2-generated holonomy group. Geometriae Dedicata. (doi:10.1007/s10711-018-0409-3).

Record type: Article

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

Identifiers

Local EPrints ID: 426624
URI: https://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: 13 Mar 2019 17:45

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