Crystallographic Helly Groups

I will use an asymptotic cone argument to show that if a
crystallographic group is Helly then its point group preserves an
L^{\infinity} metric on \R^n.  In particular, the 3-3-3 Coxeter group
is not Helly.  More generally, this result applies to virtually
abelian Helly groups, as does the converse, thus giving a full
classification of virtually abelian Helly groups.