Which finitely generated Abelian groups admit isomorphic Cayley graphs?

TL;DR

This paper characterizes when two finitely generated Abelian groups have isomorphic Cayley graphs, showing it depends on their rank and torsion part cardinality, using elementary geometric arguments.

Contribution

It provides a complete classification of finitely generated Abelian groups with isomorphic Cayley graphs based on rank and torsion size, using elementary methods.

Findings

Cayley graphs of finitely generated Abelian groups are rigid.

Isomorphic Cayley graphs imply equal rank and torsion size.

The classification uses elementary geometric arguments.

Abstract

We show that Cayley graphs of finitely generated Abelian groups are rather rigid. As a consequence we obtain that two finitely generated Abelian groups admit isomorphic Cayley graphs if and only if they have the same rank and their torsion parts have the same cardinality. The proof uses only elementary arguments and is formulated in a geometric language.