Simple Group Actions on Arc-Transitive Graphs with Prescribed Transitive Local Action

Abstract

This paper gives a partial answer to a question asked by Pierre-Emmanuel Caprace at the Groups St Andrews conference at Birmingham (UK) in August 2017, and investigated at the ‘Tutte Centenary Retreat’ workshop held at MATRIX in November 2017. Caprace asked if there exists a 2-transitive permutation group P such that only finitely many simple groups act arc-transitively on a connected graph X with local action P (of the stabiliser of a vertex v on the neighbourhood of v). Some evidence is given to suggest that the answer is “No”, even when ‘2-transitive’ is replaced by ‘transitive’, and then by way of illustration, a follow-up question is answered by showing that all but finitely many alternating groups have such an action on a 6-valent connected graph with vertex-stabiliser A6.