Highly arc transitive digraphs: reachability, topological groups

Let D be a locally finite, connected, 1-arc transitive digraph. It is shown that the reachability relation is not universal in D provided that the stabilizer of an edge satisfies certain conditions which seem to be typical for highly arc transitive digraphs. As an implication, the reachability relation cannot be universal in highly arc transitive digraphs with prime in- or out-degree. fferent aspects of the connection between highly arc transitive digraphs and the theory of totally disconnected locally compact groups are also considered.