Towards a Stallings-type theorem for finite groups

A recent development in graph-minor theory is to study local separators, vertex-sets that separate graphs locally but not necessarily globally. The local separators of a graph roughly correspond to the genuine separators of its local covering: a usually infinite graph obtained by keeping all local structure of the original graph while unfolding all other structure as much as possible. We use local separators and local coverings to discover and prove a low-order Stallings-type result for finite nilpotent groups $\mathrm{\Gamma }$: the $r$-local covering of some Cayley graph $G$ of $\mathrm{\Gamma }$ has $\ge 2$ ends that are separated by $\le 2$ vertices iff $G$ has an $r$-local separator of size $\le 2$ and $\mathrm{\Gamma }$ has order $>r$, iff $\mathrm{\Gamma }$ is isomorphic to $C_i\times C_j$ for some $i>r$ and $j\in \{1,2\}$.