I would like to know what are the applications of the theory of $n$-dimensional crystallographic groups (aka space groups, aka Bieberbach groups)
in mathematics
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").
