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 Gσ 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