GAG 2019
Compatible local actions in arc-transitive digraphs
Given a G-arc-transitive graph, the local action is the permutation group induced by a vertex-stabiliser Gv on the neighbourhood of v. It plays an extremely important role in the study of these objects.
The situation for digraphs is a little more complicated. One can consider both the in- and out-local-action, that is, the permutation group induced by a vertex-stabiliser on the corresponding in- and out-neighbourhood, respectively. Perhaps surprisingly, these two permutation groups need not be isomorphic, they may not even have the same order. We say that two permutation groups are compatible if they arise in this way, that is, as the in- and out-local action of some finite G-arc-transitive digraph. We will discuss the problem of characterising compatible permutation groups.
