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Journal of Lie Theory 14 (2004), No. 1, 035--068
Copyright Heldermann Verlag 2004
Metric Rigidity of Crystallographic Groups
Marcel Steiner
FHBB, Departement Industrie, Abteilung Maschinenbau, Gründenstraße 40, 4132 Muttenz,
Switzerland, marcel.steiner@fhbb.ch
Consider a finite set of Euclidean motions and ask what kind of conditions are
necessary for this set to generate a crystallographic group. We investigate a set
of Euclidean motions together with a special concept motivated by real crystalline
structures existing in nature, called an essential crystallographic set of isometries.
An essential crystallographic set of isometries can be endowed with a crystallographic
pseudogroup structure. Under certain well chosen conditions on the essential
crystallographic set of isometries Γ we show that the elements in Γ
define a crystallographic group G, and an embedding Φ from Γ to G exists
which is an almost isomorphism close to the identity map. The subset of Euclidean
motions in Γ with small rotational parts defines the lattice in the group G.
An essential crystallographic set of isometries therefore contains a very slightly
deformed part of a crystallographic group. This can be interpreted as a sort of metric
rigidity of crystallographic groups: if there is an essential crystallographic set
of isometries which is metrically close to an inner part of a crystallographic group,
then there exists a local homomorphism-preserving embedding in this crystallographic
group.
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