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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,

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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