Abstract

This paper is devoted to the conjecture saying that, for any connected locally finite graph Γ and any vertex-transitive group G of automorphisms of Γ, at least one of the following assertions holds: (1) There exists an imprimitivity system σ of G on V(Γ) with finite (maybe one-element) blocks such that the stabilizer of a vertex of the factor graph Γ/σ in the induced group of automorphisms G^σ is finite. (2) The graph Γ is hyperbolic (i.e., for some positive integer n, the graph Γⁿ defined by V(Γⁿ)=V(Γ) and E(Γⁿ)={{x,y}:0<d_Γ(x,y)≤n} contains the regular tree of valency 3). Our approach to the conjecture consists in fixing a finite permutation group R and considering the conjecture under the assumption that the stabilizer of a vertex of Γ in G induces on the neighborhood of the vertex a group permutation isomorphic to R. In the paper we elaborate a method (the modified track method) which allows us to prove the conjecture for many groups R. The paper consists of two parts. The present first part of the paper involves results on which the modified track method arguments are based, and a few first applications of the method. The second part is devoted to applications of the modified track method.

Article Outline

1. Introduction

2. The Main Conjecture and some related results

3. A trial of the local approach to the Main Conjecture

4. Background for the modified track method

5. The modified track method. First applications

References

This paper is an extended version of the author’s talk at the Mini-Workshop “Amalgams for Graphs and Geometries” held at Mathematisches Forschungsinstitut Oberwolfach in May, 2004. The visit was kindly supported by the German Science Foundation (DFG) and the association of friends of Oberwolfach (the “Förderverein”).