Free groups and ends of graphs

Let X be a connected graph. An automorphism of X is said to be parabolic if it leaves no finite subset of vertices in X invariant and fixes precisely one end of X and hyperbolic if it leaves no finite subset of vertices in X invariant and fixes precisely two ends of X. Various questions concerning dynamics of parabolic and hyperbolic automorphisms are discussed. The set of ends which are fixed by some hyperbolic element of a group G acting on X is denoted by H(G). If G contains a hyperbolic automorphism of X and G fixes no end of X, then G contains a free subgroup F such that H(F) is dense in H(G) with respect to the natural topology on the ends of X. As an application we obtain the following: A group which acts transitively on a connected graph and fixes no end has a free subgroup whose directions are dense in the end boundary.