The CI problem for infinite groups

TL;DR

This paper extends the concept of CI-groups to infinite groups, characterizes certain classes of these groups, and explores their properties and open problems in the context of Cayley graphs.

Contribution

It introduces new definitions for infinite CI-groups, characterizes locally-finite DCI-graphs on a7^n, and investigates properties and open questions about infinite (D)CI-groups.

Findings

Infinite (D)CI-groups must be torsion and not locally-finite.

Identifies infinite families of (D)CI$_f$-groups with specific properties.

Provides a complete characterization of locally-finite DCI-graphs on a7^n.

Abstract

A finite group $G$ is a DCI-group if, whenever $S$ and $S'$ are subsets of $G$ with the Cayley graphs Cay$(G,S)$ and Cay$(G,S')$ isomorphic, there exists an automorphism $\varphi$ of $G$ with $\varphi(S)=S'$. 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.