Generators of Bieberbach groups with 2-generated holonomy group

Ho Yiu Chung

doi:10.1007/s10711-018-0409-3

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.

Highlights

We first introduce the geometric definition of a crystallographic groups
A group is said to be an n-dimensional crystallographic group if it is a discrete subgroup of Rn O(n), which is the group of isomotries of Rn and it acts cocompactly on Rn
We apply Theorem A to get the desired bound for generators of the Bieberbach group

Summary

Introduction

We first introduce the geometric definition of a crystallographic groups. A group is said to be an n-dimensional crystallographic group if it is a discrete subgroup of Rn O(n), which is the group of isomotries of Rn and it acts cocompactly on Rn. Conjecture 1.1 [8, Dekimpe–Penninckx] Let be an n-dimensional Bieberbach group. Theorem A Let be an n-dimensional crystallographic group with holonomy group isomorphic to Cm = g|gm = 1 where m ≥ 3. The other corollary shows an n-dimensional Bieberbach group with a simple group as holonomy group can be generated by n − 1 elements. Theorem B Let be an n-dimensional crystallographic group with holonomy group isomorphic to a finite group G,. The idea of the proof of Theorem B is to apply results from [11] to get a relation between the number of generators of the finite group G and its Sylow p-subgroups. Theorem C Let be an n-dimensional Bieberbach group with 2-generated holonomy group. The idea of the proof of Theorem C is to consider a Bieberbach subgroup with cyclic holonomy group. We apply Theorem A to get the desired bound for generators of the Bieberbach group

Background

Generators of ZCm-module

Main result