The main purpose of this note is to construct an infinite , highly arc-transitive digraph with finite out-valency , and with out-spread greater than 1 , which does not have the two-way infinite path Z as a homomorphic image . This answers Question 3 . 8 in the paper [3] of Cameron , Praeger and Wormald . Our digraph has infinite in-valency . By a result of Praeger (see [5]) , a connected , vertex-and edge-transitive digraph with finite , but dif fering in-and out-valencies has Z as a homomorphic image . In the final section of the paper , we give a new proof of this result which emerged from discussions with Peter Cameron . The novelty of our approach here is that it uses the existence of Haar measure on the automorphism group of such a digraph . Of course , it should be stressed that Praeger's original proof of her result is completely elementary , but it is possible that our approach may have further applications . 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 .
First , for convenience , we review briefly the necessary definitions from [3] and introduce some notation .
We work throughout with digraphs (i . e . directed graphs) Ν A , 5 Ν . Thus , 5 is an anti-symmetric , irreflexive binary relation on A . If a A , then a 5 Ο Ν b A : a 5 b Ν . If s β«ήβ¬ , an s -arc is a sequence ( a 1 , . . . , a s Ο© 1 ) with a i 5 a i Ο© 1 for i Ρ s . We denote by a Γ© s the vertices of A that can be reached from a by s -arcs , and a Γ© Ο ! s β«ήβ¬ a Γ© s . The digraph is highly arc transiti e if its automorphism group is transitive on s -arcs , for all s β«ήβ¬ .
The following definition is taken from [3 , Definition 3 . 1] .