The CI problem for infinite groups

Abstract

A finite group $G$ is a DCI-group if, whenever $S$ and $S^{\prime }$ are subsets of $G$ with the Cayley graphs Cay$(G,S)$ and Cay$(G,S^{\prime })$ isomorphic, there exists an automorphism $\phi$ of $G$ with $\phi (S)=S^{\prime }$. It is a CI-group if this condition holds under the restricted assumption that $S=S^{-1}$. We extend these definitions to infinite groups, and make two closely-related definitions: an infinite group is a strongly (D)CI${}_f$-group if the same condition holds under the restricted assumption that $S$ is finite; and an infinite group is a (D)CI${}_f$-group if the same condition holds whenever $S$ is both finite and generates $G$. We prove that an infinite (D)CI-group must be a torsion group that is not locally-finite. We find infinite families of groups that are (D)CI${}_f$-groups but not strongly (D)CI${}_f$-groups, and that are strongly (D)CI${}_f$-groups but not (D)CI-groups. We discuss which of these properties are inherited by subgroups. Finally, we completely characterise the locally-finite DCI-graphs on ${\mathbb{Z}}^n$. We suggest several open problems related to these ideas, including the question of whether or not any infinite (D)CI-group exists.

17 pages