On fixity of arc-transitive graphs

The relative fixity of a permutation group is the maximum proportion of the points fixed by a non-trivial element of the group, and the relative fixity of a graph is the relative fixity of its automorphism group, viewed as a permutation group on the vertex-set of the graph. We prove in this paper that the relative fixity of connected 2-arc-transitive graphs of a fixed valence tends to 0 as the number of vertices grows to infinity. We prove the same result for the class of arc-transitive graphs of a fixed prime valence, and more generally, for any class of arc-transitive locally-L graphs, where L is a fixed quasiprimitive graph-restrictive permutation group.