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(G₁;S₁) and Cay(G₂;S₂) are connected Cayley graphs of finite valency on two nilpotent groups G₁ and G₂, then every isomorphism from Cay(G₁;S₁) to Cay(G₂;S₂) factors through to a well-defined affine map from G₁/N₁ to G₂/N₂, where N_i is the torsion subgroup of G_i. For the special case where the groups are abelian, these results were previously proved by A.A.Ryabchenko and C.Löh, respectively.

This work was partially supported by Australian Research Council grant DE130101001 and a research grant from the Natural Sciences and Engineering Research Council of Canada.