On the fixed ends of hyperbolic translations of infinite graphs

Let X be an infinite, connected, locally finite and vertex-transitive graph with infinitely many ends and let G be a subgroup of Aut(X) which acts transitively on X. In this note we provide a necessary and sufficient condition for the existence of a hyperbolic translation g in G with fixed ends in two prescribed open subsets of the space of ends of X. We also give an explicit combinatorial construction of the hyperbolic translation g in the special case where X is a (right) Cayley graph of a (non-abelian) free group of finite type G.