Vertex stabilizers of graphs and tracks, I

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 Γ n defined by V ( Γ n ) = V ( Γ ) and E ( Γ n ) = { { 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.