The action of nilpotent groups on infinite graphs (Q1057875)

First we prove a result about the action of nilpotent groups on the set of ends of locally finite graphs. This theorem has immediate consequences for the structure of graphs which allow a transitive action of those groups. Further we investigate the cycle-structure of automorphisms of a transitive nilpotent group and the existence of Abelian groups acting on set of imprimitivity of graphs whose automorphism groups have transitive nilpotent subgroups.