Abstract
In A. Malnič, D. Marušič, N. Seifter, P. Šparl and B. Zgrablič, Reachability relations in digraphs, Europ. J. Combin. 29 (2008), 1566–1581, it was shown that properties of digraphs such as growth, property Z , and number of ends are reflected by the properties of certain reachability relations defined on the vertices of the corresponding digraphs. In this paper we study these relations in connection with certain properties of automorphism groups of transitive digraphs. In particular, one of the main results shows that if a transitive digraph admits a nilpotent subgroup of automorphisms with finitely many orbits, then its nilpotency class and the number of orbits are closely related to particular properties of reachability relations defined on the digraphs in question. The obtained results have interesting implications for Cayley digraphs of certain types of groups such as torsion-free groups of polynomial growth.
Highlights
Introduction and preliminariesIn [2], highly arc-transitive digraphs were considered from several different viewpoints, leading to – besides many nice results – a number of interesting problems
In [6] it was shown that properties of digraphs such as growth, property Z, and number of ends are reflected by the properties of certain reachability relations defined on the vertices of the corresponding digraphs
One of the main results shows that if a transitive digraph admits a nilpotent subgroup of automorphisms with finitely many orbits, its nilpotency class and the number of orbits are closely related to particular properties of reachability relations defined on the digraphs in question
Summary
In [2], highly arc-transitive digraphs were considered from several different viewpoints, leading to – besides many nice results – a number of interesting problems. For a given pair of vertices u, v, the set of all such walks is denoted by Rk+[u, v]. In [6] it was shown that properties of the two sequences of equivalence relations (Rk+)k∈Z+ and (Rk−)k∈Z+ are strongly related to other properties of digraphs such as having property Z, the number of ends, growth conditions and vertex degree. We emphasize that we consider these quotient digraphs as simple digraphs in the sense that if there are several edges in the same direction between two sets in τ , the quotient digraph contains exactly one directed edge between the respective vertices These quotient graphs might contain loops if there is an edge (u, v) ∈ E(D) for some u ∈ vτ. Considering R− the first two possibilities are the same, but if D/R− is neither of these digraphs, it is a regular tree with outdegree 1 and indegree > 1
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