Transitive digraphs with more than one end

In this paper we investigate infinite, locally finite, connected, transitive digraphs with more than one end. For undirected graphs with these properties it has been shown that they are trees as soon as they are 2-arc transitive. In the case of digraphs the situation is much more involved. We show that these graphs can have both thick and thin ends, even if they are highly arc transitive. Hence they are far away from being ‘tree-like’. On the other hand all known examples of digraphs with more than one end are either highly arc transitive or at most 1-arc transitive. We conjecture that infinite, locally finite, connected, 2-arc transitive digraphs with more than one end are highly arc transitive and prove that this conjecture holds for digraphs with prime in- and out-degree and connected cuts.