Asymptotic stability of Cayley graphs on abelian groups

Abstract

For a finite group $G$, we say that a Cayley graph $\Gamma$ on $G$ is a most rigid representation (MRR) of $G$ if its full automorphism group has the smallest possible order among all Cayley graphs on $G$, and say that $\Gamma$ is stable if every automorphism of $\Gamma \times K_2$ comes from $\mathrm{Aut}(Γ)\times\Aut(K_2)$. Although the study of stability has attracted significant attention, particularly regarding Cayley graphs on abelian groups, a complete classification is currently out of reach even for Cayley graphs on cyclic groups. In this paper, we prove that almost all Cayley graphs on finite abelian groups are stable MRRs. This strengthens the main result of Dobson, Spiga and Verret [Combinatorica, 36 (2016), no.~4, 371--393], which states that almost all Cayley graphs on finite abelian groups are MRRs.