Joy Morris (University of Lethbridge, Canada)

Title: Oriented Regular Representations

Abstract:
A Graphical Regular Representation (GRR) for a group $G$ is a Cayley graph on $G$ whose full automorphism group is $G$. Similarly, a Digraphical Regular Representation (DRR) for $G$ is a Cayley digraph on $G$ whose full automorphism group is $G$. In the 1970s, a series of results showed that with some small exceptions and two infinite families of groups that have an obvious obstruction, every finite group admits a GRR. In 1980, Babai showed that with some small exceptions, every finite group admits a GRR. In the same paper, Babai discussed ``oriented Cayley graphs”: that is, Cayley digraphs Cay$(G,S)$ with the property that $S cap S^{-1}=emptyset$. He defined an Oriented Regular Representation (ORR) for $G$ to be an oriented Cayley graph on $G$ whose full automorphism group is $G$. He pointed out one infinite family of groups that have an obvious obstruction, and noted that based on previous work it was already known that with one small exception, every group of odd order admits an ORR.

Pablo Spiga and I recently completed a series of 3 papers (two together, and one by him) proving that with some small exceptions and the infinite family pointed out by Babai, every finite group admits an ORR. I will present some history and overview of this problem, and give an idea of some of the techniques used in our proofs.