Sur les atomes d'un graphe de Cayley infini. (On the atoms of an infinite Cayley graph) (Q1114715)

Let G be an infinite group and let S be a finite subset of G. The outconnectivity of the Cayley graph \(X=Cay(G,S)\) is \(\kappa^+(X)=Min\{| (FS)\setminus F|:\) F is a finite nonvoid subset of \(G\}\). A positive end is a finite subset R such that \(\kappa^+(X)=| (RS)\setminus R|\), which is minimal with respect to this property. We prove that there is a unique positive end containing 1. Moreover this end is a subgroup. As an application we deduce some properties of the connectivity which were known only in the finite case.