On quasi-transitive amenable graphs

In this note we use the term graph for simple, connected, undirected graphs with bounded degree vertices only. We will mostly consider infinite graphs. We denote the set of vertices of a graph G by V (G), we use E(G) for the set of edges. By ~ E(G) we denote the set of oriented edges: ~ E(G) = {(x, y) | {x, y} ∈ E(G)}. We denote the opposite orientation (y, x) of an oriented edge e = (x, y) by ē. We consider the Hilbert space l2(G) of the l2 functions u : ~ E(G) → IR satisfying u(ē) = −u(e) with the scalar product 〈u, u 〉 = 1/2 ∑ e∈ ~ E(G) u(e)u (e). For simplicity we write u(x, y) for the value u((x, y)). For a function v : V (G) → IR we define its differential dv : ~ E(G) → IR by dv(x, y) = v(y) − v(x). We call a v : V (G) → IR function a Dirichlet function if dv ∈ l2(G) and denote the set of Dirichlet functions by D(G). Let l2(V (G)) denote the Hilbert space of the l2 functions v : V (G) → IR (with the standard scalar product), this is clearly contained in D(G). Consider the adjoint d of the operator d : l2(V (G)) → l2(G). We call a function u ∈ l2(G) a flow if d u = 0. We call a Dirichlet function v ∈ D(G) harmonic if dv is a flow. We denote the set of harmonic Dirichlet functions by HD(G). Here du is given by du(x) = ∑ {y,x}∈E(G) u(y, x) for u ∈ l2(G) and x ∈ V (G). Thus a u is a flow if and only if ∑ {x,y}∈E(G) u(x, y) = 0 for every x ∈ V . The function v ∈ D(G) is harmonic if and only if for every x ∈ V the value v(x) is the average of the values v(y) with {x, y} ∈ E(G). All constant functions V (G) → IR are harmonic Dirichlet functions. For vertices x and y of a graph G let δ(x, y) denote their distance in G. A wobbling is a map f : V (G) → V (G) such that δ(x, f(x)) for x ∈ V (G) is bounded. The map f : V (G) → V (G) is called a quasi-isomorphism from G to G if there exits a positive number k—the distortion of f—such that for vertices x and y in V (G) one has