NON-PROPERNESS OF AMENABLE ACTIONS ON GRAPHS WITH INFINITELY MANY ENDS

Abstract

We study amenable actions on graphs having infinitely many ends, giving a generalized answer to Ceccherini’s question on groups with infinitely many ends. 1 Statement of the result An action of a group G on a set X is amenable if there exists a G-invariant mean on X, i.e. a map μ : 2 = P(X)→ [0, 1] such that μ(X) = 1, μ(A∪B) = μ(A)+μ(B), for every disjoint subsets A, B ⊆ X, and μ(gA) = μ(A), ∀g ∈ G, ∀A ⊆ X. An isometric action of a group G on a metric space (X, d) is proper if for some x0 ∈ X, and every R > 0, the set {g ∈ G | d(x0, gx0) ≤ R} is finite. The aim of this note is to give a short proof of the following result: Theorem 1. Let X = (V,E) be a locally finite graph with infinitely many ends. Let X = V ∪ ∂X be the end compactification. Let G be a group of automorphisms of X. Assume that the action of G on V is amenable and there exists x0 ∈ V such that the orbit Gx0 is dense in X. Then there is a unique G-fixed end in ∂X, and the action of G (as a discrete group) on V is not proper. A deep result of Stallings [4] says that G has infinitely many ends if and only if G is an amalgamated free product Γ1 ∗AΓ2 or HNN -extension HNN(Γ, A, φ) with A finite (with min{[Γ1 : A], [Γ2 : A]} ≥ 2, not both 2, in the amalgamated product case; and min{[Γ : A], [Γ : φ(A)]} ≥ 2, not both 2, in the HNN case). In particular, if G has infinitely many ends, it contains non-abelian free subgroups, hence is non amenable. Tullio Ceccherini-Silberstein asked whether non-amenability of G could be proved without appealing to Stallings’ theorem. Since a finitely generated group G with infinitely many ends acts properly and transitively on its Cayley graph, our result shows that G is not amenable.