Descendants in infinite, primitive, highly arc-transitive digraphs

The descendant set desc ( α ) of a vertex α in a directed graph (digraph) is the subdigraph on the set of vertices reachable by a directed path from α . We investigate desc ( α ) in an infinite highly arc-transitive digraph D with finite out-valency and whose automorphism group is vertex-primitive. We formulate three conditions which the subdigraph desc ( α ) must satisfy and show that a digraph Γ satisfying our conditions is constructed in a particular way from a certain bipartite digraph Σ , which we think of as its ‘building block’. In particular, Γ has infinitely many ends. Moreover, we construct a family of infinite (imprimitive) highly arc-transitive digraphs whose descendant sets satisfy our conditions and are not trees.