On the Action of Nilpotent and Metabelian Groups on Infinite, Locally Finite Graphs

In [8] J. Tits proved that a solvable fixed point free automorphism group of a tree T leaves invariant an end or a pair of ends of T. In this paper we prove the same assertion for nilpotent groups which contain certain automorphisms of infinite order and locally finite, infinite graphs. Further, we show that a metabelian group G also leaves invariant an end or a pair of ends of a locally finite graph if G satisfies special assumptions. As corollaries we obtain results about the structure of graphs which allow the transitive action of the above-mentioned groups