Applications of n-dimensional crystallographic groups I would like to know what are the applications of the theory of $n$-dimensional crystallographic groups (aka space groups) 
1) in mathematics
2) outside of mathematics,
besides the applications to $2$-dimensional and $3$-dimensional crystallography (or related fields like chemistry, or physics, of crystals).
One possible application of $4$-dimensional space groups is already reported in the wikipedia article I linked to (see "Magnetic groups and time reversal").
 A: They occur as cusps cross-sections of non-uniform hyperbolic lattices of one higher dimension. For example, they are useful in the classification of minimal volume lattices. 
A: They are used in string theory to construct Conformal Field Theories which describe orbifold limits of Calabi-Yau spaces. See for example Dixon, Harvey, Vafa and Witten, "Strings on Orbifolds I,II" Nucl. Phys. B274 (19860 285 and Nucl. Phys. B261 (1985) 678 for an early application in string theory and Miles Reid in http://arxiv.org/pdf/math/9911165v1.pdf for a more mathematical take on related material.
A: The following are applications in the theory of $p$-groups: 
Space groups have been used by 


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*Felsch, Neubüser, Plesken: Space groups and groups of prime-power order. IV: Counterexamples to the class-breadth conjecture. Journal London Math. Soc. (2), 24 (1981) 113-122 


to construct counterexamples to the class-breadth conjecture for $p=2$. Recall the the conjecture claims $\text{class} \le \text{breath} + 1$ for $p$-groups $P$ where the breath $b$ is defined such that $p^b$ is the maximal size of the conjugacy classes of $P$. In their counterexamples $P=S/2^kT$ where $S$ is a space group, $T$ the translation subgroup and $k$ a carefully choosen integer. 
Space groups those point groups are $p$-groups are also the core in proving the celebrated coclass conjectures of Leedham-Green and Newman (see the book Leedham-Green, McKay: The structure of groups of prime power order, 2002). I don't know enough to tell details, but it's striking that the series of papers that contain the proof are titled "Space groups and groups of prime-power order" (I-VIII). 
A: One of the well-known applications of crystallographic groups is  the classification
of flat complete Riemannian manifolds by their fundamental group, which is
a  torsion-free crystallographic group (aka Bieberbach group). A very nice book
about this is  "Spaces of constant Curvature" by Joseph A. Wolf.
There are many interesting generalizations in this direction. One is due to
John Milnor and Louis Auslander, so  called affine crystallographic groups. 
Here the Bieberbach theorems for crystallographic groups have been generalized, at least 
conjecturally. Every (Euclidean) crystallographic group is virtually abelan (the translations forming 
an abelian normal subgroup of finite index).
The generalization to affine crystallographic groups should be that such groups are virtually
polycyclic. In othe rwords, the fundamental group of a conplete compact affine manifold should
be virtually polycyclic. This is still an open conjecture, called Auslander's conjecture.
It has received a lot of attention, see the work of Abels, Margulis and Soifer, ranging from 1995 until 2014 (and perhaps longer).
