Bieberbach group

A Bieberbach group , named after Ludwig Bieberbach , is a space group in group theory , i.e. a discrete group of isometries of Euclidean space with a limited fundamental domain , with the additional property that every point of Euclidean space is only stabilized by the identity operation . These room groups are also called free of fixed points .

Definitions

We consider the group of isometrics of Euclidean space, that is, the one with the Euclidean metric. (See Movement (mathematics) #Movements in Euclidean space .) Some of these movements (e.g. rotations ) stabilize one or more points in Euclidean space, others (e.g. displacements) leave no point fixed. ${\displaystyle E(n)}$ ${\displaystyle \mathbb {R} ^{n}}$

A space group (or crystallographic group) is a discrete group of isometries of Euclidean space with a limited fundamental domain. Here is a group discreetly if there are no consequences to exist. The boundedness of the fundamental domain is equivalent to the conditions that (or equivalently ) is compact . ${\displaystyle \Gamma \subset E(n)}$ ${\displaystyle id\not =\gamma _{n}\in \Gamma }$ ${\displaystyle \gamma _{n}\to id}$ ${\displaystyle \Gamma \backslash \mathbb {R} ^{n}}$ ${\displaystyle \Gamma \backslash E(n)}$

In crystallography, three-dimensional groups of rooms play an essential role.

A Bieberbach group is a space group whose effect on the fixed point is free. That means there is no point with ${\displaystyle \Gamma }$ ${\displaystyle \mathbb {R} ^{n}}$ ${\displaystyle x\in \mathbb {R} ^{n}}$

${\displaystyle \gamma .x=x\ \forall \gamma \in \Gamma }$

gives. Note that individual elements can have fixed points, so a Bieberbach group can also contain reflections or rotations, because only the existence of a common fixed point is excluded. ${\displaystyle \Gamma }$ ${\displaystyle \gamma \in \Gamma }$

The fixed-point-free effect is equivalent to the condition that from and for one always follows. Another equivalent condition is that the quotient space is a manifold (not just an orbifold ). Because there is a group of isometrics, then the flat metric of the "inherits" . Bieberbach groups are precisely the fundamental groups of compact flat manifolds , which is why they are also of interest in differential geometry . ${\displaystyle \gamma \in \Gamma }$ ${\displaystyle \gamma ^{n}=id}$ ${\displaystyle n\geq 1}$ ${\displaystyle \gamma =id}$ ${\displaystyle M=\Gamma \backslash \mathbb {R} ^{n}}$ ${\displaystyle \Gamma }$ ${\displaystyle M}$ ${\displaystyle \mathbb {R} ^{n}}$

Examples

Section of a grid. The blue dots belong to the grid. Every parallelogram forms a fundamental domain.

An example of a Bieberbach group is the group of "integer shifts" of the plane. This consists of the through

${\displaystyle t_{m,n}\colon \mathbb {R} ^{2}\to \mathbb {R} ^{2},\ \ t_{m,n}(x,y):=(x+m,y+n)}$

defined shifts, where and run through all integers. This group is isomorphic to , it is discrete and a limited fundamental domain is given, for example, by the unit square . The group operates free of fixed points because all elements (except identity) are displacements. ${\displaystyle m}$ ${\displaystyle n}$ ${\displaystyle \mathbb {Z} ^{2}}$ ${\displaystyle \left\{(x,y)\in \mathbb {R} ^{2}\colon 0\leq x,y\leq 1\right\}}$

More generally, you can use any grid (as in the picture on the right), the shifts parallel to the vectors of this grid then define a 2-dimensional Bieberbach group, which of course is also too isomorphic in the abstract and which has one of the parallelograms as a compact fundamental domain. But there are also 2-dimensional Bieberbach groups in which, in addition to shifts, there is also a reflection, see Klein bottle . ${\displaystyle \mathbb {Z} ^{2}}$

These examples can be generalized to higher dimensions, for each n-dimensional grid in the the shifts along the grid define a Bieberbach group. In higher dimensions there are numerous other Bieberbach groups (see flat manifold ) for example 10 different isomorphism types of three-dimensional Bieberbach groups. (On the other hand, Bieberbach groups form only a small part of all space groups, for example, according to the Schoenflies-Fjodorow classification, there are 230 isomorphism types of three-dimensional space groups.) ${\displaystyle \mathbb {R} ^{n}}$

See also

• Space group
• Bieberbach's theorems
• Flat manifold

literature

• Ludwig Bieberbach: About the movement groups of the Euclidean spaces (first treatise) . In: Mathematical Annals . tape 70 , no. 3 , September 1911, p. 297–336 , doi : 10.1007 / BF01564500 ( gdz.sub.uni-goettingen.de ).
• Ludwig Bieberbach: About the movement groups of the Euclidean spaces. (Second treatise.) The groups with a finite fundamental domain . In: Mathematical Annals . tape 72 , no. 3 , September 1912, p. 400-412 , doi : 10.1007 / BF01456724 ( gdz.sub.uni-goettingen.de ).
• Leonard S. Charlap: Bieberbach groups and flat manifolds. Springer, New York NY et al. 1986, ISBN 3-540-96395-2 .
• Erhard Scholz : symmetry, group, duality. On the relationship between theoretical mathematics and applications in crystallography and structural engineering of the 19th century (=  Science Networks. Historical Studies . Volume 1 ). Birkhäuser, Basel et al. 1989, ISBN 3-7643-1974-7 , p. 151 ff . (At the same time: Wuppertal, comprehensive university, habilitation paper, 1986, general on Bieberbach and the Bieberbach sentences in his occupation with room groups).