×

The Cayley graphs of \(\mathbb Z^d\) and the limits of vertex-primitive graphs of \(HA\)-type. (Russian, English) Zbl 1164.05379

Sib. Mat. Zh. 48, No. 3, 606-620 (2007); translation in Sib. Math. J. 48, No. 3, 489-499 (2007).
Summary: We study the limits of the finite graphs that admit some vertex-primitive automorphism group with a regular abelian normal subgroup. It was shown in [M. Giudici, C. H. Li, Ch. E. Praeger, A. Seress, and V. I. Trofimov, J. Comb. Theory, Ser. A 114, No. 1, 110–134 (2007; Zbl 1106.05046)] that these limits are Cayley graphs of the groups \(\mathbb Z^d\). In this article we prove that for each \(d > 1\) the set of Cayley graphs of \(\mathbb Z^d\) 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 \(\mathbb Z^d\) that are limits of minimal vertex-primitive graphs. The proofs rely on a connection of the automorphism groups of Cayley graphs of \(\mathbb Z^d\) with crystallographic groups.

MSC:

05C25 Graphs and abstract algebra (groups, rings, fields, etc.)
20H15 Other geometric groups, including crystallographic groups
20K15 Torsion-free groups, finite rank

Citations:

Zbl 1106.05046
PDFBibTeX XMLCite
Full Text: EuDML EMIS