Non-treeability for product group actions

Greg Hjorth^{∗}

July 17, 2006

Abstract

A product of two lcsc non-compact groups G_{1} × G_{2} acting freely by measure preserving transformation on a standard Borel probability space gives rise to a non-treeable equivalence relation unless both groups are amenable.

The result of the present paper can be placed as one part in a series.

Theorem 0.1 (Adams, [1]) Let E_{1}, E_{2} be countably infinite measure preserving equivalence relations on standard Borel probability spaces X_{1}, X_{2} with E_{1} ×E_{2} non-amenable. Then E_{1} ×E_{2} is non-treeable.

Theorem 0.2 (Gaboriau, [11]) Let G_{1}, G_{2} be countably infinite groups with G_{1} × G_{2} nonamenable. Then any measure preserving action of G_{1} ×G_{2} on a standard Borel probability space is non-treeable.

Theorem 0.3 (Kechris, [18]) Let G_{1}, G_{2} be locally compact non-compact second countable groups one of which contains a non-compact closed amenable subgroup and with G_{1} × G_{2} nonamenable. Then any measure preserving action of G_{1} ×G_{2} on a standard Borel probability space is non-treeable.

Theorem 0.4 (Pemantle, Peres, [26]) Let X and Y be infinite, locally finite graphs. Suppose that G ⊂ Aut(X) is a closed nonamenable subgroup of Aut(X), and H ⊂ Aut(Y) has an infinite orbit. Then there is no G × H-invariant probability measure on the set of spanning trees of the direct product graph G × H.

While here we show:

Theorem 0.5 Let G_{1}, G_{2} be locally compact second countable groups with G_{1} × G_{2} acting in a measure preserving, measurable manner on a standard Borel probability space with the orbit equivalence relation E_{G_{1}} non-amenable and G_{2} not acting essentially transitively on any of its ergodic componenents. Then the orbit equivalence relation induced by the product group action, E_{G_{1}×G_{2}}, is not treeable.

Corollary 0.6 Let G_{1}, G_{2} be locally compact non-compact second countable groups with G_{1}×G_{2} acting in a free, measure preserving, measurable manner on a standard Borel probability space. Then if G_{1} × G_{2} is non-amenable then E_{G_{1}×G_{2}} is not treeable.

∗Partially supported by NSF grant DMS 0140503

Our proof is a refinement of Kechris’, which in turn had points in common with Adams’, and as thus traces back to ideas from Zimmer, [31], and from there to earlier work of Margulis and Mostow. The original proof of Gaboriau’s result, however, used a radically different idea to these three results, and the proof of Pemantle and Peres was different again. It is clear that 0.5 implies 0.3 and 0.2; in the last section we use some observations from [20] to show that 0.4 follows as well, but this is less of an achievement since their proof is by far the simplest and most elegant of the sequence. Again as pointed out by [20], 0.4 can be used to derive 0.2, though it should also be said that 0.2 arises as one small piece of a much broader body of work on the concept of cost. 0.5 also implies 0.1, since any countable measure preserving equivalence relation is induced by the measure preserving action of a countable group ([9]).

Acknowledgments: I am grateful to Nicholas Monod for a number of helpful conversations around the topic of the Poisson boundary, and indeed for suggesting it might have some applications to the theory of Borel reducibility. I am also grateful to Alexander Kechris for pointing out the relevance of §29, 30 from [20]. I am grateful to Scot Adams for a number of discussions about these techiques in general, as well as some specific comments about this specific paper. And finally I am grateful to the referee for an unusually insightful reading; indeed the final statement of 0.5 is sharper than the original version and borrows directly from a suggestion of the referee.

1 Borel reducibility

In this paper we only consider actions by groups which are locally compact, second countable, and Hausdorff – lcsc groups for short. Any such group is completely metrizable (see for example [24], chapter 3, exercise 3). Thus in particular it is Polish, and there is a robust structure theorem for its Borel sets and Borel actions (see for instance [5], [16]).

We will find it convenient to organize the later arguments around the notion of Borel reducibility.

Definition Let E, F be Borel equivalence relations on standard Borel spaces X, Y. We say that E is Borel reducible to F, written E ≤_{B} F if there is a Borel function

θ : X → Y

such that for all x_{1}, x_{2} ∈ X

x_{1}Ex_{2} ⇔ θ(x_{1})F θ(x_{2}).

We say that E is smooth if it is Borel reducible to id(ℝ), the identity relation on ℝ.

Definition An equivalence relation E on standard Borel X is treeable if there is a Borel subset G ⊂ X × X such that:

(i) G is symmetric;

(ii) G is irreflexive, in the sense that (x, x) is never in G;

(iii) G is acyclic (in other words, if x_{1}, x_{2}, ..., x_{n} ∈ G with each (x_{i}, x_{i+1}) ∈ G and x_{i} ≠ x_{j} for i ≠ j, then (x_{n}, x_{1}) ∉ G); (iv) the connected components of G form the equivalence classes of E.

An equivalence relation is countable if every equivalence relation is countable.

It has been customary to only consider treeability in the context of countable Borel equivalence relations, but it makes sense to consider treeable equivalence relations with uncountable classes and certain key aspects of the existing theory simply adapts.

Lemma 1.1 (essentially [14]) Let E on X be a treeable Borel equivalence relation and let A ⊂ X be Borel set meeting each orbit in at most a countable set. Then E|_{A}, the restriction of E to A, is again treeable.

Proof Let B be the set of points which are either in A or lie between two distinct points in A. By Novikov-Lusin (see [16]), this set is Borel, and it is clear that E|_{B} is treeable. Now [14] shows that the restriction of a countable treeable equivalence relation to a Borel subset is again treeable. □

Corollary 1.2 Let G be lcsc group acting in a Borel manner on a standard Borel space X. Then E_{G}^{X} is treeable if and only if E_{G}^{X} is Borel reducible to an orbit equivalence relation induced by a free, Borel action of 𝔽_{2}, the free group on two generators.

Proof Following [17], we can fix Borel set A ⊂ X meeting each orbit in a countable non-empty set. By the uniformization theorem for Borel sets in the plane with countable sections^{1} we can in a Borel manner select to each x ∈ X some x′ = ϕ(x) ∈ [x]_{G} ∩ A, and hence obtain E_{G}^{X} ≤_{B} E_{G}^{X}|_{A}.

(⇒:) It suffices to show that E_{G}^{X}|_{A} is Borel reducible to a suitable action of 𝔽_{2}. But by the last lemma E_{G}^{X}|_{A} is again treeable, and then since it is countable we have that it is Borel reducible to a free, Borel action of 𝔽_{2} by [14].

(⇐:) Assume θ : X → Y gives a reduction of E_{G}^{X} to E_{𝔽_{2}}^{Y}, where Y is a free, Borel 𝔽_{2} space. Then in particular θ|_{A} reduces E_{G}^{X}|_{A} to a treeable equivalence relation, and hence, by [14] again, E_{G}^{X}|_{A} admits some treeing, G_{A}. We then define a treeing by xGx′ if and only if either:

(i) x, x′ ∈ A, xG_{A}x′; or

(ii) x ∈ A, x′ ∉ A, ϕ(x′) = x; or
(iii) x ∈ A, x′ ∉ A, ϕ(x) = x′.
□

The treeable relations are most naturally contrasted with the hyperfinite.

Definition An equivalence relation E is hyperfinite if there is an increasing union of Borel equivalence relations, (F_{n}), with finite classes, such that

E = ⋃_{n∈ℕ}F_{n}.

Hyperfinite equivalence relations can be alternately characterized as the equivalence relations induced by a Borel action of ℤ or the Borel equivalence relations for which every infinite equivalence class can be assigned, in a uniformly Borel fashion, the structure of a ℤ-chain.^{2} This last characterization makes it clear that hyperfiniteness is a special kind of treeability.

However whereas most of the structural properties of hyperfinitess are understood, at least up to discounting null sets, the structural properties of the treeable equivalence relations have not been adequately documented.

Question Is a countable Borel equivalence relation with finite index over treeable again treeable?

See here [14] for a discussion of this problem and terminology. In that paper they do show that finite index over hyperfinite is again hyperfinite.

Question Which lcsc groups have all their free Borel, measure preserving actions on standard Borel probability spaces treeable?

In the case of hyperfinite this is largely understood: If every such action of lcsc G is Borel reducible to a hyperfinite equivalence relation then G is amenable (see e.g. [31]), and conversely if G is amenable then after possibly discounting a null set any such action will be Borel reducible to a hyperfinite equivalence relation (see [31], [25]). (It remains notoriously open, however, even for discrete groups, whether any such action will be hyperfinite everywhere, and not just almost everywhere.)

1Again, [16] is an ideal place to find an account of classical facts such as this one.

2See [22] for an extended analysis of some of the subtle distinctions which are possible here.

On the other hand, there is at least one structural property known true for hyperfinite and known false for treeable: Increasing unions of hyperfinite equivalence relations are hyperfinite almost everywhere by [7]; an increasing union of countable treeable equivalence relations need not be treeable on any conull set (see [14] or [11]).

An especially important example of a hyperfinite equivalence relation is given by E_{0}.

Notation We let E_{0} be the equivalence relation of eventual agreement on infinite binary sequences. So for x⃗, y⃗ ∈ {0, 1}^{ℕ}(=_{df} 2^{ℕ}) we let

x⃗E_{0}y⃗

if ∃N∀m > N(x_{m} = y_{m}).

E_{0} is clearly hyperfinite; conversely [8] shows that every hyperfinite equivalence relation is Borel reducible to E_{0}.

Notation We let E_{1} be the equivalence relation of eventual agreement on infinite sequences of reals: For x⃗, y⃗ ∈ ℝ^{ℕ} we let

x⃗E_{1}y⃗

if ∃N∀m > N(x_{m} = y_{m}).

E_{1} is an example of a treeable equivalence relation which is not Borel reducible to any countable Borel equivalence relation. In fact by [19], not Borel reducible to any Borel action of a Polish group.

Notation We let E_{∞T} arise from the orbit equivalence relation of 𝔽_{2} on F (2^{𝔽_{2}}), the part of 2^{𝔽_{2}}(=_{df} {0, 1}^{𝔽_{2}}) on which every non-trivial element of 𝔽_{2} has no fixed points under the shift action.

For our purposes the exact definition of E_{∞T} is unimportant. We only need the fact established in [14] that it is induced by a free action of 𝔽_{2} and that any countable, Borel, treeable equivalence relation is Borel reducible to E_{∞T}. With this notation granted, I can finish this brief survey with one last, and, for the author, most central question:

Question Up to ≤_{B}-reducibility, how many Borel equivalence relations E are there with

E_{0} <_{B} E < E_{∞T}?

2 Mixing properties

Definition A locally compact group G equipped with a Haar measure ν is said to act measurably on a standard Borel probability space (X, μ) if for any Borel A ⊂ X the set {(g, x) : g · x ∈ A} is measurable in (G × X, ν × μ). Let G be a lcsc group acting in a properly ergodic, measurable, non-singular manner on a standard Borel probability space (X, μ). The action of G on (X, μ) is mildly mixing if whenever G acts in a properly ergodic, measurable, non-singular, manner on a standard Borel probability space (X′, μ′) the diagonal action of G on (X × X′, μ × μ′) is again ergodic. ([28])

For M a class of separable Banach spaces, the action of G on (X, μ) is M-ergodic if whenever E ∈ M and we have a continuous homomorphism

π : G → Iso(E),
g ↦ π_{g},

from G to the isometry group of E, and we have a measurable

φ : X → E

which is equivariant in the sense that

φ(g · x) = π_{g}(φ(x))

all x ∈ X, g ∈ G, then φ is constant almost everywhere.

A couple of remarks about these definitions.

There is essentially only one standard Borel structure on a separable Banach space, and that is the Borel structure arising from the Polish topology induced by the Banach norm. This is the sense in which we require φ : X → E to be measurable.

Since the isometry group of separable Banach space is Polish, it follows from Pettis’ lemma, [27], that any measurable homomorphism G → Isom(E) is necessarily continuous.

Future discussions of M-ergodicity will probably be clarified by noting that a standard argument shows that an a.e. equivariant map from X to E can be replaced by an everywhere equivariant map.

Lemma 2.1 Let G be a locally compact group equipped with right invariant Haar measure acting in a non-singular measurable manner on standard Borel probability space (X, μ). Let π : G → Isom(E) be a continuous homomorphism and φ̂ : X → E a measurable map with a.e. x ∈ X, g ∈ G

φ̂(g · x) = π_{g}(φ̂(x)).

Then we can find conull, invariant X_{0} ⊂ X and φ : X_{0} → E agreeing a.e. with φ̂ and such that for all x ∈ X_{0}, g ∈ G

φ(g · x) = π_{g}(φ(x)).

Proof Let A ⊂ G × X be the set of (g, x) with φ̂(g · x) = π_{g}(φ̂(x)). We have assumed that A is conull, and then in particular we have that for any h ∈ G the set of (g, x) with (g, h · x) ∈ A is again conull. Then by Fubini the set

X_{0} = {x ∈ X : a.e. g, h ∈ G((g, h · x) ∈ A)}

is again conull; by right invariance X_{0} is G-invariant.

Given any x ∈ X_{0} we choose some h ∈ G such that for a.e. g ∈ G we have (g, h · x) ∈ A and let φ(x) = π_{h}^{−1}(φ̂(h · x)); this definition does not depend on the choice of h, since given a competing choice h′ we may find g_{1}, g_{2} such that g_{1}h = g_{2}h′ and (g_{1}, h · x), (g_{2}, h′ · x) ∈ A, and note that

π_{h}^{−1}(φ̂(h · x)) = π_{h}^{−1}π_{g_{1}}^{−1}(φ̂(g_{1}h · x))
= π_{h′}^{−1}π_{g_{2}}^{−1}(φ̂(g_{2}h′ · x)) = π_{h′}^{−1}(φ̂(h′ · x)).

A similar calculation gives that φ is equivariant on all of X_{0}. □

A parallel and well known argument gives that a measurable action of a lcsc group G on a standard Borel probability space (X, μ) agrees a.e. with a Borel action of G on X (see [31]). We will generally only assume our actions are measurable, but in light of this well known argument it is reasonable to blur the distinction between Borel actions and merely measurable actions.

Lemma 2.2 Mild mixing implies M-ergodicity for M the class of all separable Banach spaces.

Proof Given π : G → Isom(E) and equivariant f : X → E, we obtain a G-invariant function

X × X → ℝ
(x, y) ↦ ||f(x) − f(y)||.

If the action of G on X × X is ergodic, which is the case assuming the original action was mild mixing, then this function must be constant a.e., which would imply it is zero a.e. □

The converse to this lemma fails. There is an action of 𝔽_{2} which is M-ergodic for M the class of all separable Banach spaces but is not mildly mixing. This counterexample uses extraneous ideas, and so we put it off until the end of §3.

Nevertheless, M-ergodicity does imply the analog of mild mixing if we restrict to taking products with measure preserving actions. Here we provide a definition.

Definition A properly ergodic, measurable, non-singular action of a lcsc group G on (X, μ) is moderately mixing if whenever G acts in a properly ergodic, measurable, measure preserving manner on a standard Borel probability space (X′, μ′) the diagonal action of G on (X×X′, μ×μ′) is again ergodic.

Lemma 2.3 Let G be a lcsc group acting in a measurable, non-singular, ergodic manner on a standard Borel probability space (X, μ). The action of G on (X, μ) is moderately mixing if and only if it is {ℓ^{2}}-ergodic.

Proof (⇐) : Suppose G acts by measure preserving transformations on a standard Borel probability space (Y, ν). For a contradiction assume A ⊂ X ×Y is measurable, invariant, and neither null nor conull. At each x ∈ X we let A_{x} = {y ∈ Y : (x, y) ∈ A}. Define

π : X → L^{2}(Y)
x ↦ χ_{A_{x}},

the function which assigns to each x the characteristic function of the slice A_{x}. This is a.e. G-equivariant, and so the assumptions on X imply it is constant a.e. as an element in L^{2}(Y). Thus we have some A ⊂ Y such that A_{x} = A a.e. x, and in particular for every g ∈ G the measure of AΔg · A is zero. Replacing A by {y ∈ Y : a.e. g ∈ G(g · y ∈ A)}, gives a truly invariant subset of Y, with a contradiction to ergodicity.

(⇒): Assume

π : G → LinIsom(ℓ^{2})
g ↦ π_{g}

is a continuous homomorphism and we have π-equivariant measurable ϕ : X → ℓ^{2} with ϕ(g·x) = π_{g} · ϕ(x) a.e. x, g. We assume π is not constant and try to construct a counterexample to moderate mixing.

Claim: There is an irreducible unitary representation

ρ : G → LinIsom(H)
g ↦ ρ_{g}

and non-trivial measurable ψ : X → H with ψ(g · x) = ρ_{g} · ϕ(x) a.e. x, g.

Proof of Claim: Following say [21] we find a standard measure space (Z, m) and write ℓ^{2} as a direct integral of irreducible representations of G:

ℓ^{2} = ∫_{Z} H_{z}dz

ϕ = ∫_{Z} ϕ^{z}dz.

Then for each z ∈ Z we have a map

p_{z} : ℓ^{2} → H_{z}

which coordinate so that for any v ∈ ℓ^{2} we have

v = ∫_{Z} p_{z}(v)dz

and for any g ∈ G

g · v = ∫_{Z} ϕ_{g}^{z} · p_{z}(v)dz.

Thus for a.e. g, x we have at m-a.e. z

p_{z}(ϕ_{g} · π(x)) = ϕ_{g}^{z} · p_{z}(π(x));

we may assume that p_{z} ∘ π is non-trivial at every z. Thus we may appeal to Fubini to find a single z such that for a.e. g, x p_{z}(ϕ_{g} · π(x)) = ϕ_{g}^{z} · p_{z}(π(x)) and finish with H = H_{z}, ρ = ϕ^{z}, ψ = p_{z}(π). (Claim□)

Now there is a split in cases:

Case 1: The representation is finite dimensional.

Clearly there is a U(H)-invariant invariant probability measure ν′ on the unit ball in H. For each x ∈ X we let (Y_{x}, ν_{x}) be the ergodic component of ψ(x) in ((H)_{1}, ν′) under the action of G. By ergodicity of the G action on X, we obtain that this ergodic component is constant μ-a.e; let (Y, μ) be this ergodic component.

Now we obtain a non-trivial G-invariant function

f : X × Y → ℝ
(x, v) ↦ ||ψ(x) − v||.

Case 2: The representation is infinite dimensional.

Following [31] 5.2.13 we may find a standard Borel probability space (Y, ν) on which G acts measurably, measure preservingly, and ergodically, and for which the representation ρ is isomorphic to a direct summand of the usual representation of G on L^{2}(Y, ν); at that point we may very well assume H is a direct summand of L^{2}(Y, ν), and hence every f ∈ H is a function in L^{2}(Y, ν).

We obtain a non-trivial function with

f : X × Y → ℂ
(x, y) ↦ (ψ(x))(y).

Unwinding the definitions we see that a.e.

f(g · x, g · y) = (ψ(g · x))(g · y)
= g · (ψ(x))(g · y) = ψ(x)(g^{−1}g · y) = ψ(x)(y),

since for any function F ∈ L^{2}(Y, ν) we have by definition that g · F (z) = F (g^{−1}z) any z ∈ Y. □

Schmidt and Walters have obtained a purely combinatorial characterization of mild mixing:

Theorem 2.4 ([28]) Let G act in a measurable, non-singular, properly ergodic manner on (X, μ). The action is mild mixing if and only if for all A measurable and neither null nor conull

liminf_{g→∞}μ(AΔ(g · A)) > 0.

It would be interesting if there is a similar combinatorial characterization of when a properly ergodic non-singular action preserves ergodicity with respect to products with measure preserving, ergodic actions on standard Borel probability spaces.

We work towards a general result, due to Kaimanovich, and after Burger and Monod. In this section we see any locally compact group has an M-ergodic action, for M the class of separable Banach spaces. In the next section we review the amenability properties of this action.

Definition Let μ be a probability measure on a lcsc group G that has the same measure class as a left invariant Haar measure. Consider the one sided product,

∏_{n∈ℕ}(G, μ),

equipped with the measure λ arising from the infinite product of μ. We equip this space with the shift action

T : ∏_{n∈ℕ}G → ∏_{n∈ℕ}G

(g_{1}, g_{2}, g_{3}, ...) ↦ (g_{2}, g_{3}, g_{4}, ...).

Lemma 2.5 (Kaimanovich; [15]) If E is a separable Banach space and

F : ∏_{ℕ}G → E

π : G → Iso(E)

are measurable with

π_{g_{1}}F ((g_{2}, g_{3}, ...)) = F ((g_{1}, g_{2}, ...)),

then F is constant a.e.

Note that we do not yet assume π is a homomorphism, though in actual fact it will be in the cases of interest to us. The measurability assumption on F is that pullbacks of sets in E which are open in the Banach space metric are λ measurable.

We need to define another semigroup action along with another action of the group G.

Definition Let

S : ∏_{n∈ℕ}G → ∏_{n∈ℕ}G

(g_{1}, g_{2}, g_{3}, ...) ↦ (g_{1}g_{2}, g_{3}, g_{4}, ...).

Note that the assumptions on μ guarantee that this semigroup action is non-singular with respect to the product measure λ.

We then let G act on ∏_{ℕ} G by

γ · (g_{1}, g_{2}, g_{3}, ...) = (γg_{1}, g_{2}, g_{3}, ...).

This G action clearly commutes with S and is again non-singular by choice of μ.

Definition Let Γ(G, μ) be the space of ergodic components of S; that is to say, we write ∏_{ℕ} G as a measurable union of disjoint pieces,

∏_{ℕ}G = ˙ ⋃_{x∈Γ(G,μ)}Y_{x},

where each Y_{x} is S-invariant and S acts ergodically on Y_{x}. (See [12]). We then let

b : ∏ G ↠ Γ(G, μ)

be the natural surjection, with b(g⃗) = x if g⃗ ∈ Y_{x}. We let ν = b^{∗}[λ] be the push forward of λ, defined by ν[A] = λ[b^{−1}[A]].

Since the action of G commutes with S we obtain a well defined action of G on Γ(G, μ) with

γ · b(g⃗) = b(γ · g⃗).

Again this action is non-singular, since the pre-existing action on (∏_{N} G, λ) is non-singular.

Therefore we have a daunting total of four non-singular actions: The semigroup action provided by T; the semigroup action provided by S; the action on the original ∏_{ℕ} G by G; the induced action of G on the quotient object Γ(G, μ). Ultimately we will only be interested in this final action of G, but it is the remarkable discovery of [15] that the other three actions cooperate to prove:

Theorem 2.6 (Kaimanovich, [15]) Let M be the class of separable Banach spaces. Then the action of G on Γ(G, μ) is M-ergodic.

Proof (In any reasonable mathematical sense, this proof has been previously given in [15]; however since our set up and notation is rather different, we recall it here for the reader’s convenience.) Fix E ∈ M and π : G → Iso(E) a continuous homomorphism. Let f : Γ(G, μ) → E be G-equivariant and measurable in the usual sense. We then define F = f ∘ b

F : ∏_{ℕ}G → E,

g⃗ ↦ f(b(g⃗)).

F is clearly measurable; on the other hand we obtain a.e. that

π_{g_{1}}F ((g_{2}, g_{3}, ...)) = π_{g_{1}}(f(b((g_{2}, g_{3}, ...)))
= f(g_{1} · b((g_{2}, g_{3}, ...))),

by the G-equivariance of f,

= f(b(g_{1} · (g_{2}, g_{3}, ...))),

by definition of the G-action on Γ(G, μ),

= f(b(g_{1}g_{2}, g_{3}, ...)) = f(b(g_{1}, g_{2}, g_{3}, ...)) = F ((g_{1}, g_{2}, g_{3}, ...)),

by invariance of b under S; hence by 2.5 f is constant a.e. □

By suitably composing any g⃗ ∈ ∏_{ℕ} G we can view such a sequence as providing a random walk through G. The shift action S then corresponds to discounting the choice of the first element in this walk. The space of ergodic components Γ(G, μ) is something like the space of random walks considered up to “direction” or “eventual behavior”. We have failed to emphasize this point of view above; it comes through much more strongly in the original treatment of [15].

Kaimanovich’s paper also proves a better result: The G-action on Γ(G, μ) is doubly ergodic. That is to say, any measurable, G-equivariant function f : Γ(G, μ) × Γ(G, μ) → E is constant a.e. The proof of this stronger result, for which we have no need, is similar and follows from a strengthening of 2.5.

3 Amenability

Definition (See [31]; but note that, unlike Zimmer, we write our actions on the left.) Let G be a lcsc group acting measurably on (X, μ) and let H be a Polish group. A measurable function

α : X × G → H

is a cocycle if for a.e. x ∈ X and all g, h ∈ G

α(x, gh) = α(hx, g)α(x, h).

In the case that H = Iso(E), the isometry group of some separable Banach space, we obtain the dual cocycle

α^{∗} : X × G → Iso(E^{∗})

by taking the adjoint in the usual way.

Definition We say that the action of G on (X, μ) is amenable if whenever we have such a cocycle and a measurable assignment

x ↦ A_{x},
X → K_{c}(E_{1}^{∗}),

assigning to each point a weak^{∗} compact, convex subset of the unit ball E^{∗} with the invariance property

α^{∗}(x, g)A_{x} = A_{gx}

a.e. x, all g, then we can find a measurable assignment

X → E^{∗}
x ↦ a_{x},

with each a_{x} ∈ A_{x} and α^{∗}(x, g)a_{x} = a_{g·x} a.e. x, all g.

We are thinking of E as having the weak^{∗} topology, and so in particular measurability of the assignment x ↦ A_{x} is the requirement that it be measurable with respect to the induced Effros Borel structure: That is to say, if U ⊂ E^{∗} is weak^{∗} open, then the set of x with A_{x} ∩ U ≠ ∅ must be μ-measurable.

There is also a parallel definition we can give for amenability of equivalence relations. We do not give this definition here, but simply quote an alternative characterization from [7].

Theorem 3.1 (Connes-Feldman-Weiss; [7]) An equivalence relation E on a standard Borel probabibility space is amenable if and only if there is a conull subset A with E|_{A} ≤_{B} E_{0}.

Lemma 3.2 (Zimmer; [31]) If S is an amenable G-space and X is any standard Borel probability space on which G acts measurably and non-singularly, then X × S is an amenable G-space in the usual diagonal action.

Theorem 3.3 (Zimmer; [30]) If G is lcsc, then Γ(G, μ), as defined in §2, is an amenable G-space.

Putting together 3.2 and 3.3 along with 2.6 we get:

Corollary 3.4 An lcsc group G admits a non-singular, measurable action on a standard Borel probability space (S, m) such that for any other standard Borel probability space (X, ν) on which G acts measurably and in a measure preserving, properly ergodic fashion, we have:

(i) the diagonal action of G on (S × X, m × ν) is properly ergodic; (ii) the diagonal action of G on (S × X, m × ν) is amenable.

Without placing additional restrictions on G, this corollary would seem to be the strongest we can hope for in this direction. In particular, we cannot have a mildly mixing action of 𝔽_{2} which is amenable.

Proposition 3.5 (Folklore) Let the free group 𝔽_{2} on two generators act amenably on a standard Borel probability space (S, m) by non-singular transformations. Then the action of 𝔽_{2} on (S^{3}, m^{3}) is not properly ergodic.

Proof Note that 𝔽_{2} acts continuously on its boundary, ∂𝔽_{2}, and hence we have an induced homomorphism ρ : 𝔽_{2} → IsoC(∂𝔽_{2}), from 𝔽_{2} to the isometry group of the continuous functions on the boundary. We then obtain an induced cocycle

S × 𝔽_{2} → Iso(C(∂𝔽_{2}))
(s, σ) ↦ ρ(σ).

The action is amenable, and hence for M(∂𝔽_{2}) the space of probability measures on ∂𝔽_{2} (and the dual of C(∂𝔽_{2})) we obtain a measurable equivariant assignment

S → M(∂𝔽_{2}),
s ↦ ν_{s}.

At the level of S^{3} we obtain a corresponding equivariant

S^{3} → M(∂𝔽_{2}),

(s_{1}, s_{2}, s_{3}) ↦ \frac{1}{3} (ν_{s_{1}} + ν_{s_{2}} + ν_{s_{3}}).

On an invariant non-null set, A ⊂ S^{3}, the resulting measure ν_{s_{1}} +ν_{s_{2}} +ν_{s_{3}} does not concentrate on just two points. Following say §C.4 [13] (or see [2] and [4] for the primary references) we may find a measurable equivariant map from the measures on M(∂𝔽_{2}) concentrating on more than two points, and hence there is a measurable equivariant map

φ : A → 𝔽_{2}.

This gives us a selector A_{0} ⊂ A defined by

A_{0} = {(s_{1}, s_{2}, s_{3}) ∈ A : φ(s_{1}, s_{2}, s_{3}) = e},

where e is the identity in 𝔽_{2}; the existence of such a selector contradicts proper ergodicity of 𝔽_{2} on A and hence on S^{3}. □

There is one last fact in this direction we need.

Lemma 3.6 If lcsc G acts on standard Borel probability space (X, μ) by measure preserving transformations with every point having an amenable stabilizer, then if E_{G}^{X} ≤_{B} E_{0}, then the group G is amenable.

Proof From say 2.15 [14] we have that the equivalence relation is amenable; and then it is known from [3] that an action with amenable stabilizers and amenable equivalence relation is amenable, and thus G is amenable by 4.3.3[31]. □

4 The result

Theorem 4.1 Let G_{1}, G_{2} be lcsc groups acting on a standard Borel probability space (X, μ) with:

(0) the actions of G_{1} and G_{2} commute;

(i) the action of G_{1} × G_{2} is measurable (as a function from G_{1} × G_{2} × X to X); (ii) the action of G_{1} × G_{2} is measure preserving; (iii) the orbit equivalence relation E_{G_{1}} is not amenable; (iv) the action of G_{2} is properly ergodic on all its ergodic components.

Then the resulting orbit equivalence relation E_{G_{1}×G_{2}}^{X} is non-treeable.

Proof By going to ergodic components we may actually assume that the action of G_{1} × G_{2} is ergodic. Assume for a contradiction that the equivalence relation is treeable, and then appealing to 1.2 we obtain that some Borel θ : X → F (2^{𝔽_{2}}) witnessing E_{G_{1}×G_{2}}^{X} Borel reducible to the free Borel action of 𝔽_{2} on the standard Borel probability space F (2^{𝔽_{2}}). We in particular obtain a corresponding cocycle

α : X × G_{1} × G_{2} → 𝔽_{2}.

Following the ergodic decomposition theorem of [12], we write

X = ˙ ⋃_{z∈Z}X_{z}

as a decomposition of X into G_{2}-invariant ergodic pieces^{3}; applying the measure disintegration theorem (see for instance [16]) we obtain a measure μ̂ on Z along with G_{2}-invariant measures (μ_{z})_{z∈Z} such that μ = ∫_{Z} μ_{z}μ̂(z). The assumptions of the theorem imply that the action of G_{2} on each X_{z} is properly ergodic. This is a critical observation for the proof – if G_{2} were to act essentially transitively on its ergodic components, then it would make no real contribution to the complexity of the structure of the equivalence relation, the argument in the first claim below would fail, and in fact the theorem without this assumption would admit a counterexample.

Now appealing to 3.4 we let (S, m) be a standard Borel probability space on which G_{2} acts ergodically with the diagonal action on X × S amenable and such that at each fiber X_{z} we have the induced action of G_{2} on X_{z} × S still properly ergodic. Let

α̂ : (X × S) × G_{2} → 𝔽_{2}

be derived from α in the obvious way:

α̂(x, s, g) = α(x, e_{G_{1}}, g),

where e_{G_{1}} is the identity in G_{1}. By the amenability of the action of G_{2} on X × S we may find a measurable assignment of probability measures

X × S → M(∂𝔽_{2})
(x, s) ↦ ν_{x,s}

such that α̂(x, s, g) · ν_{x,s} = ν_{g·x,g·s} a.e. x ∈ X, s ∈ S all g ∈ G_{2}.

We let M_{≤2}(∂𝔽_{2}) be the probability measures concentrating on at most two points; we let M_{≥3}(∂𝔽_{2}) = M(∂𝔽_{2}) \ M_{≤2}(∂𝔽_{2}) be the measures whose support is greater than two points.

The next argument through the next couple of claim parallels [1] and is similar to ideas of Zimmer, and from there Margulis and eventually Mostow. Claim: For a.e. x, s, ν_{x,s} ∈ M_{≤2}(∂𝔽_{2}). Proof of claim: Otherwise we can obtain some ergodic component X_{z} such that for a.e. x ∈ X_{z} and a.e. s ∈ S we have ν_{x,s} ∈ M_{≥3}(∂𝔽_{2}). Appealing to the existence of an equivariant Borel map from M_{≥3}(∂𝔽_{2}) to 𝔽_{2}, as given discussed in the course of 3.5 (again see for instance theorem C4 of [13], or [2], [4]) we may find a measurable α̂-equivariant φ : X_{z} × S → 𝔽_{2}.

This will provide a contradiction to proper ergodicity of the action. We let (σ_{i})_{i∈ℕ} enumerate 𝔽_{2} and at each (x, s) ∈ X_{z} × S we let i(x, s) be the least i for which there exists a non-null collection of g ∈ G_{2} with φ(g · x, g · s) = σ_{i}; since the measure quantifier preserves Borel sets (see for instance [16]), this function is Borel, and moreover the set

A_{1} = {(x, s) : φ(x, s) = σ_{i(x,s)}}

will provide a Borel selector for E_{G_{2}}^{X_{z}×S}; since α̂(x, s, g) · θ(x) = θ(g · x) by the definition of the cocycle, we have

θ(x_{1}) = θ(x_{2})

for all (x_{1}, s_{1}), (x_{2}, s_{2}) ∈ A_{1} with (x_{1}, s_{1})E_{G_{2}}^{X_{z}×S}(x_{2}, s_{2}); then using say Jankov, von Neumann uniformization (see for instance [16]) we obtain a measurable function f : X_{z} × S → F (2^{𝔽_{2}}) assigning to each (x, s) the unique value θ(x′) any (x′, s′) ∈ A_{1} that is E_{G_{2}} equivalent to (x, s). Our action is ergodic, and f is G_{2}-invariant, so f is constant a.e. But that means in particular that the equivalence class [θ(x)]_{𝔽_{2}} is constant almost everywhere, and hence, by assumption on θ, the equivalence class [x]_{G_{2}} is constant almost everywhere on X_{z}. This contradicts the proper ergodicity of the action of G_{2} on X_{z} × S. (Claim□)

3Here Z is the standard Borel space arising as equivalence classes of the smooth equivalence relation x_{1}Ex_{2} if and only if ∀B ∈ B_{0}(a.e. g ∈ G(g · x_{1} ∈ B) ⇔ a.e. g ∈ G(g · x_{2} ∈ B)), for some countable Boolean algebra B_{0} which generates the Borel sets on X

By a straightforward exhaustion argument (compare [1], [18], or even lemma 6.6 [13]) we may find a measurable, equivariant assignment

X × S → M(∂𝔽_{2})
(x, s) ↦ ν_{x,s}

which is maximal almost everywhere. In other words, if

(x, s) ↦ ν′_{x,s}

is any other measurable, equivariant assignment with

supp(ν_{x,s}) ⊂ supp(ν′_{x,s})

(where here supp(μ) denotes the support of measure μ) and α̂(x, s, g) · ν′_{x,s} = ν′_{g·x,g·s} a.e. x a.e. g, then

ν′_{x,s} = ν_{x,s}

almost everywhere. Since ν_{x,s} ∈ M_{≤2}(∂𝔽_{2}) a.e, we may further more assume that these measures assign all points in their support equal mass – which is to say, a.e. ν_{x,s} equals either δ_{a} or \frac{1}{2} (δ_{a_{1}} + δ_{a_{2}}). Claim: For g ∈ G_{1}, the assignment

(x, s) ↦ α(g · x, g, e_{G_{2}})^{−1} · ν_{g·x,s}

is again α̂-equivariant. Proof of claim: Since for any h ∈ G_{2} we have

α̂(x, s, h) · (α(g · x, g, e_{G_{2}})^{−1} · ν_{g·x,s})
= α(x, h, e_{G_{1}}) · (α(g · x, g, e_{G_{2}})^{−1} · ν_{g·x,s}),

which by the defining identity of the cocycle and the fact that the actions of G_{1} and G_{2} commute equals

α(hg · x, g, e_{G_{2}})^{−1} · (α(g · x, h, e_{G_{1}}) · ν_{g·x,s})
= α(hg · x, g, e_{G_{2}})^{−1} · ν_{hg·x,h·s},

by G_{2}-equivariance of the cocycle, which in turn by the commutativity of the G_{1}, G_{2} actions equals

= α(gh · x, g, e_{G_{2}})^{−1} · ν_{gh·x,h·s},

as required. (Claim□) Thus by maximality of the assignment

(x, s) ↦ ν_{x,s}

we have

α(g · x, g, e_{G_{2}})^{−1} · ν_{g·x,s} = ν_{x,s}

almost everywhere, and hence that (x, s) ↦ ν_{x,s} must be equivariant under the entire action of G_{1} × G_{2}. In particular we obtain that for almost every (x, s) ∈ X × S and almost every g ∈ G_{1} we have some α(x, g, e_{G_{2}}) · ν_{x,s} = ν_{g·x,s}. Using Fubini we may find some specific s_{0} ∈ S such that for a.e. g ∈ G_{1}, x ∈ X

α(x, g, e_{G_{2}}) · ν_{x,s_{0}} = ν_{g·x,s_{0}}
∴ ν_{g·x,s_{0}}E_{𝔽_{2}}^{M(∂𝔽_{2})}ν_{x,s_{0}}.

Since the equivalence relation on M(∂𝔽_{2}) induced by the action of 𝔽_{2} is hyperfinite (see for instance the appendix of [13]), we may write

E_{𝔽_{2}}^{M(∂𝔽_{2})} = ⋃_{n∈ℕ}F_{n},

where (F_{n})_{n∈ℕ} is an increasing union of Borel equivalence relation with finite classes.

Following [17] we can fix a Borel C ⊂ X meeting each E_{G_{1}}^{X} equivalence class in a countable non-empty set and a Borel function

f : X → C

with xE_{G_{1}}^{X}f(x) all x ∈ X. In particular this gives E_{G_{1}}^{X} ≤_{B} E_{G_{1}}^{X}|_{C}. We then use f to transfer the measure μ to a measure μ̂ on C with

μ̂(A) = μ(f^{−1}[A]).

Note that E_{G_{1}}^{X}|_{C} will be non-amenable with respect to μ̂ since E_{G_{1}}^{X} is non-amenable with respect to μ.

We then set

x_{1}F̂_{n}x_{2}

if x_{1}E_{G_{1}}^{X}x_{2} and ν_{x_{1},s_{0}}F_{n}ν_{x_{2},s_{0}}. Since x_{1}E_{G_{1}}^{X}x_{2} implies ν_{x_{1},s_{0}}E_{𝔽_{2}}^{M(∂𝔽_{2})}ν_{x_{2},s_{0}} we have

⋃_{n} F̂_{n} = E_{G_{1}}^{X}.

Now there are two cases, both leading to the conclusion that E_{G_{1}}^{X} is Borel reducible to E_{0} on a co-null set, and hence a contradiction to its non-amenability by 3.6. Case(1): There is an n and a μ̂ non-null set on which F̂_{n} is non-smooth. Claim: There is a μ̂ non-null set of x ∈ C for which

θ[[x]F̂_{n}],

the pointwise image of the set [x]F̂_{n} under θ, is infinite. Proof of claim: Otherwise apply Lusin-Novikov (as say found in [16]) to obtain a Borel function

ρ : θ[C] → C

such that for every y we have

θ(ρ(y)) = y.

Then ρ ∘ θ[C] meets each F̂_{n} equivalence class in a finite non-empty set. Then since Borel equivalence relations with finite classes are smooth and since F̂_{n} is Borel reducible to F̂_{n}|_{ρ∘θ[C]} we obtain that F̂_{n} is smooth, with a contradiction to case assumption. (□Claim) Claim: There is a μ̂ non-null set of x’s for which the corresponding measure ν_{x,s_{0}} has infinite stabilizer. Proof of claim: Suppose first that ν_{x,s_{0}} has finite stabilizer. Then for any other ν with νE_{𝔽_{2}}ν_{x,s_{0}} there are only finitely many σ ∈ 𝔽_{2} with σ · ν_{x,s_{0}} = ν. Thus since each F_{n} is finite we obtain {σ ∈ 𝔽_{2} : σ · ν_{x,s_{0}}F_{n}ν_{x,s_{0}}} finite, and hence θ[[x]F̂_{n}] is finite.

In conclusion we have thus obtained that x is outside the μ̂ positive set of the last claim. (□Claim)

As noted for instance in the appendix of [13], there are only countably many measures in M_{≤2}(∂𝔽_{2}) with infinite stabilizer, and hence by ergodicity we may assume there is some fixed ν such that a.e.

ν_{x,s_{0}}E_{𝔽_{2}}^{M_{≤2}(∂𝔽_{2})}ν.

Then we may find a measurable assignment

X → 𝔽_{2}
x ↦ σ_{x}

such that σ_{x} · ν_{x,s_{0}} = ν a.e. Thus if we replace the reduction θ by

θ̂: X → F (2^{𝔽_{2}}),
x ↦ σ_{x} · θ(x)

then we obtain a reduction of E_{G_{1}}^{X} a.e. into an orbit equivalence relation the induced by the stabilizer of ν, which will necessarily be cyclic abelian, and hence hyperfinite by [14], with a contradiction to non-amenability of G_{1} by 3.6. Case(2): At every n there is a conull set on which F̂_{n} is smooth.

Hence E_{G_{1}}^{X} is, on a conull set, hypersmooth – that is to say, the increasing union of smooth equivalence relations. Hence by the Kechris-Louveau dichotomy theorem of [19] we have that it is either ≥_{B} E_{1} or ≤_{B} E_{0}. The former is impossible, since [19] shows that E_{1} is not Borel reducible to any Polish group action, whilst the latter gets us into the same contradiction to the non-amenability of E_{G_{1}}. □

5 Spanning trees for product groups

Definition For X a set, T ⊂ X × X is said to be a spanning tree if it is symmetric, irreflexive, acyclic, and moreover any two distinct points in X are connected by some path in T.

We will only be interested in the case that X is countable. In this event, the collection of such spanning trees is a Borel subset of {0, 1}^{X×X} via the identification of a subset of X × X with its characteristic function. The group of all permutations of X acts on the standard Borel space of spanning trees by g · T = {(g · x_{1}, g · x_{2}) : (x_{1}, x_{2}) ∈ T }.

The following is a continuous analog of a lemma presented in the discrete case with credit to Russ Lyons by the manuscript [20].

Lemma 5.1 Let Ω be a countable set, let Sym(Ω) be the group of all permutations of Ω, and let G < Sym(Ω) be a subgroup. Suppose:

(a) G is closed in the topology of pointwise convergence;

(b) for any a ∈ Ω the subgroup G_{a}(=_{df} {g ∈ G : g · a = a}) is compact;

(c) there is a G-invariant Borel probability measure on the spanning trees on Ω. Then there is an action of G on some standard Borel probability space Y with:

(i) the action measurable (as a function from G × Y to Y);

(ii) the action measure preserving;

(iii) for any y ∈ Y the stabilizer of y in G, G_{y}, is compact;

(iv) the equivalence relation E_{G}^{Y} is treeable.

Proof Let Ω_{0} ⊂ Ω be an orbit under G. I claim that we can replace Ω by Ω_{0} by obtaining an invariant probability measure on the spanning trees on Ω_{0}. To do this, we only need to find a Borel and invariant assignment of a spanning tree on Ω_{0} to spanning trees on Ω, and this follows by the kind of argument one sees in [14]: Given a spanning tree T on Ω, we assign to each y ∈ Ω the point p_{T} (y) arising as the place of first entrance into Ω_{0} under a T -path from y to Ω_{0}; we can do this so if the path from y to p_{T} (y) passes through y′ then p_{T} (y) = p_{T} (y′); we then define T̂ on Ω_{0} by x_{1}T̂x_{2} if and only if x_{1} ≠ x_{2} and there are y_{1}, y_{2} with p_{T} (y_{1}) = x_{1}, p_{T} (y_{2}) = x_{2}, and y_{1}T y_{2}.

Thus from now on we assume that the action of G acts transitively on Ω and we fix some specific a ∈ Ω and let G_{a} be its stabilizer in G. We use T (Ω) to denote the space of spanning trees on Ω. We let μ be the G-invariant measure T (Ω).

Following [29] we let (X, ν) be a standard Borel probability space on which G acts freely. We need only show that the orbit equivalence relation arising from the action of G on y = T (Ω)×X is treeable.

Consider then the equivalence relation E_{G_{a}} on T (Ω) × X arising from the group G_{a}. The responsible group is compact and hence the equivalence relation is smooth, and thus we may find a Borel selector A ⊂ T (Ω) × X meeting each E_{G_{a}}-equivalence class in exactly one point.

We need only define the treeing on A. For (T_{1}, x_{1}), (T_{2}, x_{2}) ∈ A set (T_{1}, x_{2})G(T_{2}, x_{2}) if and only if there is g ∈ G with

g · T_{1} = T_{2},

g · x_{1} = x_{2},
(a, g(a)) ∈ T_{2}, or, equivalently, (a, g^{−1}(a)) ∈ T_{1}.

This is a Borel graphing, since the set of g will be of size at most 1, and we can appeal to the uniformization theorem for Borel sets in the plane with small sections. To check it is a treeing we need to show existence of unique paths between ((T, x), (T′, x′)) ∈ E_{G}^{Y} ∩ A × A.

We will construct a path and then go on to verify that any other G-path must arise by the same process and in fact be the same.

We choose the unique h ∈ G with h · x = x′; the assumption of E_{G} equivalence entails h · T = T′. We then form a non-self intersecting path a_{0} = a, a_{1}, a_{2}, ..., a_{n} = h(a) from a to h(a) inside the tree T. We consider the set of all g_{1} with g_{1}^{−1}(a_{0}) = a_{1}; this exactly determines g_{1} up to a left coset of G_{a}, so we can choose some such g_{1} with g_{1} · (T, x) = (T_{1}, x_{1}) ∈ A, and observe that g_{1} witnesses (T, x)G(T_{1}, x_{1}). Applying the same recipe we may choose some g_{2} with g_{2}^{−1}(a_{0}) = g_{1}(a_{2}) and (T_{2}, x_{2}) = g_{2} · (T_{1}, x_{1}) ∈ A; since g_{1}^{−1}(a_{0}) = a_{1}

g_{1}(a_{1}) = a_{0},

and from and (a_{1}, a_{2}) ∈ T we obtain

(g_{1}(a_{1}), g_{1}(a_{2})) ∈ T_{1} = g_{1} · T
∴ (a_{0}, g_{1}(a_{2})) = (a_{0}, g_{2}^{−1}(a_{0})) ∈ T_{1}
∴ (T_{1}, x_{1})G(T_{2}, x_{2}).

We continue successively, obtaining at each i < n the pair (T_{i}, x_{i}) along with g_{1}, g_{2}, ...g_{i−1}

(T_{i}, x_{i}) = g_{i}g_{i−2}...g_{1} · (T, x),
g_{i}g_{i−2}...g_{1}(a_{i}) = a_{0},

and hence

(a_{0}, g_{i}g_{i−1}...g_{1}(a_{i+1})) = (g_{i}g_{i−1}...g_{1}(a_{i}), g_{i}g_{i−1}...g_{1}(a_{i+1})) ∈ g_{i−1}g_{i−2}...g_{1} · T = T_{i}.

We then choose g_{i+1} subject to the requirement that

g_{i+1}^{−1}(a_{0}) = g_{i}g_{i−1}...g_{1}(a_{i+1})

and (T_{i+1}, x_{i+1}) =_{df} g_{i+1} · (T_{i}, x_{i}) ∈ A. This keeps the happy game playing for another round, since g_{i+1}g_{i}...g_{1}(a_{i+1}) = a_{0} and (T_{i+1}, x_{i+1})G(T_{i}, x_{i}). Eventually we end up with g = g_{n}g_{n−1}...g_{1} with the properties that g · (T, x) = (T_{n}, x_{n}) ∈ A and g^{−1} · a_{0} = a_{n}, which in particular entails G_{a}g = G_{a}h; then the assumption that A selects exactly one point from each coset will entail (T_{n}, x_{n}) = (T′, x′) as required.

Alternatively, given any other G path, (S_{0}, y_{0}) = (T, x), (S_{1}, y_{1}), ..., (S_{m}, y_{m}) = (T′, x′) we reverse the engineering above and find h_{1}, h_{2}, ...h_{m} with each h_{i} · y_{i−1} = y_{i} and then let b_{i} = h_{1}^{−1}h_{2}^{−1}...h_{i}^{−1}(a_{0}). The assumption of G-adjacency implies each

(h_{i+1}^{−1}(a_{0}), a_{0}) ∈ S_{i} = h_{i}h_{i−1}...h_{1} · T
∴ (h_{1}^{−1}h_{2}^{−1}...h_{i}^{−1}h_{i+1}^{−1}(a_{0}), h_{1}^{−1}h_{2}^{−1}...h_{i}^{−1}(a_{0}) ∈ T)
∴ (b_{i+1}, b_{i}) ∈ T.

Note however we must also end up with h_{m}h_{m−1}...h_{1} = h, and hence b_{m} = a_{n}, and in particular a, b_{1}, b_{2}, ..., b_{m} = a_{n} traces out a path in T from a to a_{n}. It then follows from T being a tree and the assumptions on our set A that either that either m = n and each a_{i} = b_{i} or there is a repeated vertex – (S_{i}, y_{i}) = (S_{j}, y_{j}) some i ≠ j ≤ n – along the way. □

This lemma granted, 0.4 follows from 4.1. In the situation they describe we have a product of lcsc automorphism groups, the first of which is assumed to be non-amenable and the second of which is implicity assumed to be non-compact. Their free action on any probability space must be non-treeable by 4.1, and hence by the lemma above there can be no invariant probability measure on the spanning trees of X × Y in the statement of 0.4.

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