Isomorphisms of Cayley graphs on nilpotent groups

D. W. Morris, J. Morris, Gabriel Verret

    Research output: Contribution to journalArticlepeer-review

    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(1); S-1) and Cay(G(2); S-2) are connected Cayley graphs of finite valency on two nilpotent groups G(1) and G(2), then every isomorphism from Cay(G(1); S-1) to Cay(G(2); S-2) factors through to a well-defined affine map from G(1)/N-1 to G(2)/N-2, where is the torsion subgroup of G,. For the special case where the groups are abelian, these results were previously proved by A. A. Ryabchenko and C. Loh, respectively.
    Original languageEnglish
    Pages (from-to)453-467
    JournalNEW YORK JOURNAL OF MATHEMATICS
    Volume22
    Publication statusPublished - 4 Jun 2016

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