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ON GENERATORS OF CRYSTALLOGRAPHIC GROUPS AND ACTIONS ON FLAT ORBIFOLDS
 

Summary: ON GENERATORS OF CRYSTALLOGRAPHIC GROUPS
AND ACTIONS ON FLAT ORBIFOLDS
A. ADEM
, K. DEKIMPE
, N. PETROSYAN
, AND B. PUTRYCZ
Abstract. We find new bounds on the minimal number of generators
of crystallographic groups with p-group holonomy. We also show that
similar bounds exist on the minimal number of generators of the abelian-
izations of arbitrary crystallographic groups. As a consequence, we show
that this restricts the rank of elementary abelian p-groups that can act
effectively on closed connected flat orbifolds.
1. Introduction
A closed connected flat n-orbifold M is a quotient of Rn by a cocompact
action of a discrete subgroup of isometries of Rn. A group admitting
such an action is known as a crystallographic group and it is the orbifold
fundamental group of M. When is torsion-free, it is called a Bieberbach
group and it is the fundamental group of the corresponding flat manifold. By
the first Bieberbach theorem (see [2]), every n-dimensional crystallographic
group has a normal subgroup T of translations which is a lattice of Rn and

  

Source: Adem, Alejandro - Department of Mathematics, University of British Columbia

 

Collections: Mathematics