Groups acting on graphs with polynomial growth

In the first part of this paper we consider nilpotent groups G acting with finitely many orbits on infinite connected locally finite graphs X thereby showing that all α ϵ G of infinite order are automorphisms of type 2 of X. In the second part we investigate the automorphism groups of connected locally finite transitive graphs X with polynomial growth thereby showing that AUT(X) is countable if and only if it is finitely generated and nilpotent-by-finite. In this case we also prove that X is contractible to a Cayley graph C(G, H) of a nilpotent group G (for some finite generating set H) which has the same growth degree as X. If X is a transitive strip we show that AUT(X) is uncountable if and only if it contains a finitely generated metabelian subgroup with exponential growth.