Metric Rigidity of Crystallographic Groups

Marcel Steiner

Abstract: 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 $\Gamma$ we show that the elements in $\Gamma$ define a crystallographic group $G$, and an embedding $\Phi\colon\Gamma\to G$ exists which is an almost isomorphism close to the identity map. The subset of Euclidean motions in $\Gamma$ 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.

Classification (MSC2000): 20H15

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