  Let S be a finite generating set of a torsion-free, nilpotent group G. We
show that every automorphism of the Cayley graph Cay(G;S) is affine. (That is,
every automorphism of the graph is obtained by composing a group automorphism
with multiplication by an element of the group.) More generally, we show that
if Cay(G;S) and Cay(G';S') are connected Cayley graphs of finite valency on two
nilpotent groups G and G', then every isomorphism from Cay(G;S) to Cay(G';S')
factors through to a well-defined affine map from G/N to G'/N', where N and N'
are the torsion subgroups of G and G', respectively. For the special case where
the groups are abelian, these results were previously proved by A.A.Ryabchenko
and C.Loeh, respectively.
