Metric Rigidity of Crystallographic Groups

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: Γ→ 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. 1. Crystallographic Groups and (Almost) Flat Manifolds Many substances in their solid phase are crystallised. They are either monocrystals (rock crystal, sugar crystal), or have a micro-crystalline structure, i.e., they are made up of thousands of tiny mono-crystals (steel, lump of sugar). Crystalline structures are very regular. Most of the conceptual tools for the classification of crystalline structures, the theory of lattices and space groups, had been developed by the nineteenth century. In 1830 J. F. C. Hessel determined the 32 geometric classes of point groups in three-dimensional Euclidean space. In 1850 A. Bravais derived 14 types of three-dimensional lattices. C. Jordan in 1867 listed 174 types of groups of motions, including both crystallographic and non-discrete groups. The symmetry groups of crystalline structures in three-space were found independently by E. S. Fedorov in 1885 and A. Schoenflies in 1891. The determination of all crystalline structures in three-space enabled the modern definition of a crystallographic group to be formulated. Every discrete group of motions of n-dimensional Euclidean space for which the closure of the fundamental ISSN 0949–5932 / $2.50 c © Heldermann Verlag