Isomorphisms of Cayley graphs on nilpotent groups

TL;DR

This paper proves that automorphisms of Cayley graphs on torsion-free nilpotent groups are affine, and extends this to isomorphisms between such graphs on nilpotent groups, generalizing previous results for abelian groups.

Contribution

It establishes that all automorphisms of Cayley graphs on torsion-free nilpotent groups are affine and characterizes isomorphisms between such graphs, extending known results beyond abelian groups.

Findings

Automorphisms of Cay(G;S) are affine for torsion-free nilpotent groups.

Isomorphisms between Cayley graphs on nilpotent groups factor through affine maps.

Results generalize previous findings for abelian groups.

Abstract

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.