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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").

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    $\begingroup$ It depends on what you mean by "application": I could argue that theory of semisimple Lie group and symmetric spaces is application of affine Coxeter groups since these are equivalent to root systems. $\endgroup$
    – Misha
    Commented Mar 10, 2013 at 22:38
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    $\begingroup$ This is more of an application of Bieberbach's theorems: in the proof of quasi-isometric rigidity of $\mathbb{Z}^n$ given in that paper: arxiv.org/abs/math/0509527 it is proved that a group $G$ quasi-isometric to $\mathbb{Z}^n$ admits a proper isometric action on some (finite-dimensional) Euclidean space. By Bieberbach, the group $G$ is virtually abelian, so contains a $\mathbb{Z}^m$ of finite index. Finally $m=n$ by invariance of growth under quasi-isometry. $\endgroup$
    – Alain Valette
    Commented Mar 29, 2013 at 21:13

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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).

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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.

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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.

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  • $\begingroup$ Some references would make your answer more useful. $\endgroup$
    – Ian Agol
    Commented Mar 11, 2013 at 17:56
  • $\begingroup$ I added a few early references. $\endgroup$
    – Jeff Harvey
    Commented Mar 11, 2013 at 19:59
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The following are applications in the theory of $p$-groups:

Space groups have been used by

  • 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).

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