Lethbridge Number Theory and Combinatorics Seminar: Gabriel Verret

A permutation group G on X is transitive if for every x and y in X, there exists g in G mapping x to y. The group G is called primitive if, in addition, it preserves no nontrivial partition of X. Let Gamma be a vertex-primitive digraph, that is, its automorphism group acts primitively on its vertex-set. It is not hard to see that, in this case, Gamma cannot have two distinct vertices with equal neighbourhoods, unless Gamma is in some sense trivial. I will discuss some recent results about the case when Gamma has two vertices with “almost” equal neighbourhoods, and how these results were used to answer a question of Araújo and Cameron about synchronising groups. (This is joint work with Pablo Spiga.)