An infinite highly arc-transitive digraph (Q1357656)

A digraph is highly arc-transitive if its automorphism group is transitive on \(s\)-arcs, for all natural numbers \(s\). In this paper, an infinite, highly arc-transitive digraph with infinite in-valency, finite out-valency, and with out-spread greater than 1, which does not have the two-way infinite path \(Z\) as a homomorphic image, is constructed. The question of whether there is an infinite, highly arc-transitive digraph with finite (and equal) in- and out-valencies, with out-spread greater than 1, and not having \(Z\) as a homomorphic image remains open.