On the action of a group on a graph

Abstract

The present paper was written at the request of one of the editors for a survey on some results of the author. The research presents an approach to studying groups acting on connected vertex-symmetric graphs of finite valency. This situation arises naturally when studying groups (the action of a group on its Cayley graph, the action of a primitive group on the corresponding permutation graphs, etc.) and also when studying certain applications (description of graphs with given symmetry properties and some other problems of algebraic combinatorics, crystallography, etc.). The paper consists of two parts. In Part I, the asymptotic properties of automorphisms of connected vertex-symmetric graphs of finite valency are investigated. A kind of structure theory for groups acting as automorphism groups on such graphs is developed. Part II is devoted to two applications. The first one is an affirmative answer to the L. van den Dries-A. Wilkie question on the existence of a non-simply connected space associated with a finitely generated group. The second application is a proof of an improved version of the C. Sims conjecture for primitive permutation groups. Comments (1)-(7) at the end of the paper are mainly of an illustrative nature.