Isomorphisms of Cayley graphs on nilpotent 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.
Cite
@article{arxiv.1603.01883,
title = {Isomorphisms of Cayley graphs on nilpotent groups},
author = {Dave Witte Morris and Joy Morris and Gabriel Verret},
journal= {arXiv preprint arXiv:1603.01883},
year = {2016}
}
Comments
12 pages, plus 7 pages of notes to aid the referee. One of our corollaries is already known, so a reference to the literature has been added