Descendants in infinite, primitive, highly arc-transitive digraphs

Some constructions of highly arc-transitive digraphs

“…The general structure that descendant sets Γ can have in primitive highly arc-transitive digraphs was explored in [1]. In particular, it is shown that if the out-valency m of Γ is prime, then Γ must be a tree.…”

“…Our main result in this paper shows that at least some of the structures described in [1] do arise as the descendant set of an infinite primitive highly arc-transitive digraph (in addition to those which are directed trees). The construction is based on [5].…”

“…The construction is based on [5]. Moreover, following [4], we also show that the structures from [1] arise as the descendant set of (imprimitive) highly arc-transitive digraphs without property Z, where this means that there is no homomorphism onto the natural digraph on Z. (These are also known to have universal reachability relation.)…”

“…Now suppose in addition that D is infinite, primitive and highly arctransitive, with finite out-valency m, and let x ∈ V D. In [1] we investigate the subdigraph induced on desc(x), which we write as desc(x) , and which we also call the descendant subdigraph. We formulate three conditions, P1, P2 and P3, that desc(x) must satisfy and show that a digraph Γ satisfying the three conditions is constructed in a particular way from a certain bipartite digraph Σ, which we think of as its 'building block'.…”

“…It is then straightforward to construct an infinite highly arc-transitive digraph with a digraph homomorphism onto an infinite regular tree and which has Γ (Σ) as its descendant subdigraph (as was already done in [1]). In Section 4 we construct a highly arc-transitive digraph without property Z and with Γ (Σ) as it set of descendants, which is however imprimitive.…”