The Cayley graphs of ℤd and the limits of vertex-primitive graphs of HA-type

Abstract

We study the limits of the finite graphs that admit some vertex-primitive group of automorphisms with a regular abelian normal subgroup. It was shown in [1] that these limits are Cayley graphs of the groups ℤd. In this article we prove that for each d > 1 the set of Cayley graphs of ℤd presenting the limits of finite graphs with vertex-primitive and edge-transitive groups of automorphisms is countable (in fact, we explicitly give countable subsets of these limit graphs). In addition, for d < 4 we list all Cayley graphs of ℤd that are limits of minimal vertex-primitive graphs. The proofs rely on a connection of the automorphism groups of Cayley graphs of ℤd with crystallographic groups.