Justin Tatch Moore

Hindman’s Theorem, Ellis’s Lemma, and Thompson’s group F

Abstract. The purpose of this article is to formulate conjectural generalizations of Hindman’s Theorem and Ellis’s Lemma for nonassociative binary systems and relate them to the amenability problem for Thompson’s group F. Partial results are obtained for both conjectures. The paper will also contain some general analysis of the conjectures.

Mathematics Subject Classification (2010): 03E02, 03E50, 05D10, 05C55, 20F38, 43A07

Keywords: Amenable, Ellis’s Lemma, finitely additive measure, idempotent measure, Hindman’s Theorem, Thompson’s group HINDMAN’S THEOREM, ELLIS’S LEMMA, AND THOMPSON’S

GROUP F

Contents

1. Introduction 2 2. Preliminaries 5 3. A partition theorem concerning sums of at most two elements of 𝔸 7 4. Conjecture 1.4 implies Conjecture 1.3 9 5. Monotonicity properties of F -invariant measures 11 6. The set of idempotents in 𝔸_{U} 13 7. Concluding remarks 16 References 16

1. Introduction

In [17] a connection was established between the amenability of discrete groups and structural Ramsey theory. The purpose of this article is to examine, from this perspective, the problem of whether Richard Thompson’s group F is amenable. Specifically, it will be demonstrated that the question of whether F is amenable is closely related to the generalization of Hindman’s Theorem and Ellis’s Lemma to nonassociative binary systems.

Recall that a group G is amenable if there is a finitely additive (left) translation invariant probability measure μ which measures all subsets of G. Probably the most famous example of a group whose amenability is unknown is Richard Thompson’s group F. The question of its amenability was considered by R. Thompson himself [24] but was rediscovered and popularized by Geoghegan in 1979; it first appeared in the published literature in [6, p. 549].

The motivation for this question stems from the fact that F does not contain a copy of 𝔽_{2}, the free group on two generators [2]. It was a longstanding open problem of von Neumann to determine whether every nonamenable group contains a copy of 𝔽_{2} (it is easily demonstrated that any discrete group which contains 𝔽_{2} is nonamenable). A finitely generated counterexample was constructed by Ol′shanskii [19] and a finitely presented example was constructed only more recently by Ol′shanskii and Sapir [20]. Very recently, Monod constructed a new example of nonamenable group not containing 𝔽_{2} which is closely related to Thompson’s group F [15]; see also [14] for a finitely presented nonamenable subgroup of Monod’s group. Thus the original motivation for considering whether F is amenable is no longer valid. In the meantime, however, the problem of F ’s amenability took on a life of its own, owing to the fact that it is simple to define and serves as an important example in group theory. In this paper we will see that it is related to a natural problem in Ramsey theory, giving the problem renewed motivation.

The research presented in this paper was partially supported by NSF grants DMS–0757507 and DMS–1262019. I would like to thank Stevo Todorcevic and the math department at the University of Paris VII for their hospitality during my visit in July 2011. The present article underwent considerable revisions during that period. I would also like to thank Miodrag Sokič for reading an early draft of this paper and offering a number of suggestions.

Building on work of I. Schur and R. Rado and confirming a conjecture of Graham and Rothschild [8], Hindman proved the following result.

Hindman’s Theorem. [11] If c : ℕ → k is a coloring of ℕ with k colors, then there is an infinite X ⊆ ℕ such that c is monochromatic on the sums of finite subsets of X.

Hindman’s original proof of this theorem was elementary and combinatorial but quite complex. Galvin and Glazer later gave a simple proof using topological dynamics, which I will now describe (see [12, p. 102-103], [25, p. 30-33]). The operation of addition on ℕ can be extended to its Čech-Stone compactification βℕ to yield a compact left topological semigroup. Galvin realized that the existence of an idempotent U in (βℕ, +) allowed for a simple recursive construction of infinite monochromatic sets as in the conclusion of Hindman’s Theorem. Glazer then observed that the existence of such idempotents follows immediately from the following lemma of Ellis.

Ellis’s Lemma. [5] If (S, ⋆) is a compact left topological semigroup, then S contains an idempotent.

We will now examine to what extent both Hindman’s Theorem and Ellis’s Lemma can be generalized to a nonassociative setting. Let (𝕋, , 1) denote the free binary system on one generator. The algebra (𝕋, , 1) can be represented in the following manner which will be useful later when defining our model for Thompson’s group F. If a and b are subsets of (0, 1], define

a b = \frac{1}{2}a ∪ \frac{1}{2}(b + 1).

Observe that, as a function, is injective. Consequently, the binary system generated by 1 = {1} is free; we will take this as our model of 𝕋. Just as in the case of addition on ℕ, a binary operation ⋆ on a set S can be extended to βS as follows: W is in U ⋆ V if and only if

{u ∈ S : {v ∈ S : u ⋆ v ∈ W } ∈ V} ∈ U

Let us first observe that (β𝕋, ) does not contain an idempotent and that Hindman’s theorem is false if we replace ℕ by 𝕋. To see this, define l on 𝕋 recursively by:

l(1) = 0 and l(a b) = l(a) + 1.

If U is an ultrafilter on 𝕋, then

{t ∈ 𝕋 : l(t) is even} ∈ U ⇔ {t ∈ 𝕋 : l(t) is odd} ∈ U U

and in particular, U U ≠ U for any U ∈ β𝕋. Similarly, if a, b ∈ 𝕋, then l(a) and l(a b) have different parity and hence the naive generalization of Hindman’s theorem to (𝕋, ) fails. Similarly, if a, b, c ∈ 𝕋, l((a b) c) and l(a (b c)) have a different parity.

It is informative to compare this situation to a reformulation of Hindman’s Theorem.

Theorem 1.1. If c : FIN → k is a coloring of FIN with k colors, then there is an infinite sequence x_{0} << x_{1} << … of elements of FIN such that c is monochromatic on all finite unions of members of this sequence.

Here FIN denotes the nonempty finite subsets of ℕ and x << y abbreviates max(x) < min(y). This can be regarded as the corrected form of the following false statement: If c : FIN → k, then there is an infinite set X such that c is monochromatic on all nonempty finite subsets of X. The reason this statement is false is that every infinite subset of ℕ contains finite nonempty subsets of both even and odd cardinalities. Observe that this statement is equivalent to the modification of Theorem 1.1 where we require x_{i} to be a singleton for all i. Thus we can avoid this trivial counterexample by allowing singletons to be “glued” together into blocks.

For the nonassociative analog of Hindman’s Theorem, I propose a different form of “gluing.” Define 𝕋_{n} to be all elements of 𝕋 of cardinality n. These correspond to the ways to associate a sum of n ones. In particular, each 𝕋_{n} is finite and in fact the cardinalities of these sets are given by the Catalan numbers. Let 𝔸_{n} denote the collection of all probability measures on 𝕋_{n}. Notice that 𝔸_{n} can be viewed as a convex subset of the vector space generated by 𝕋_{n} and 𝕋_{n} can be regarded as the set of extreme points of 𝔸_{n}. In particular, if c : 𝕋_{n} → ℝ is any function, then c extends linearly to a function which maps 𝔸_{n} into ℝ; such extensions will be taken without further mention. Define 𝔸 to be the (disjoint) union of the sets 𝔸_{n} and define # : 𝔸 → ℕ by #(ν) = n if ν ∈ 𝔸_{n} (note that while each 𝔸_{n} is convex, 𝔸 is not). The operation of on 𝕋 extends bilinearly to a function defined on 𝔸:

μ ν(E) = ∑ ; a b∈E μ({a})ν({b})

Observe that # is a homomorphism from (𝔸, ) to (ℕ, +). A sequence μ_{i} (i < ∞) of elements of 𝔸 is increasing if i < j implies #(μ_{i}) < #(μ_{j}).

In this paper, I will prove the following partial extension of Hindman’s theorem to (𝔸, ).

Theorem 1.2. If c : 𝕋 → [0, 1] and ϵ > 0, then there is an r ∈ [0, 1] and an increasing sequence μ_{i} (i < ∞) of elements of 𝔸 such that for all i, |c(μ_{i}) − r| < ϵ and all i < j, |c(μ_{i} μ_{j}) − r| < ϵ.

In order to state the nonassociative form of Hindman’s theorem we will need the nonassociative analog of a finite sum. If t is in 𝕋_{m}, then t defines a function from 𝕋^{m} → 𝕋 by substitution: t(u_{0}, …, u_{m−1}) is obtained by simultaneously substituting u_{i} for the i^{th} occurrence of 1 in the term corresponding to t. This operation extends to an m-multilinear function which maps 𝔸^{m} into 𝔸. We are now ready to state the conjectured generalization of Hindman’s Theorem.

Conjecture 1.3. If c : 𝕋 → [0, 1] and ϵ > 0, then there is an r ∈ [0, 1] and an increasing sequence μ_{i} (i < ∞) of elements of 𝔸 such that whenever t is in 𝕋_{m} and i_{0} < … < i_{m−1} is admissible for t

|c(t(μ_{i_{0}}, …, μ_{i_{m−1}})) − r| < ϵ

Admissibility is a technical condition which will be defined later. For now it is sufficient to mention two of its properties. First, if m − 1 ⩽ i_{0} < … < i_{m−1}, then i_{0} < … < i_{m−1} is admissible for any element of 𝕋_{m}. Additionally any increasing sequence of m integers is admissible for some element of 𝕋_{m}. In particular, if c(t) is required to depend only on #(t), then the above conjecture reduces to Hindman’s Theorem.

It was shown in [17] that even a weak form of Hindman’s theorem for (𝔸, ) is sufficient to prove that Thompson’s group F is amenable. The relationship between F ’s amenability and the Ramsey theory of (𝔸, ) becomes even more apparent when one attempts to generalize Ellis’s Lemma. The extension of a binary operation ⋆ on a set S mentioned above can be generalized so as to extend ⋆ to the space ℓ^{∞}(S)^{∗}, which contains the set Pr(S) of all finitely additive probability measures on S:

μ ⋆ ν(f) = ∫ ∫ f(x ⋆ y)dν(y)dμ(x)

This leads to the following conjectural extension of Ellis’s Lemma.

Conjecture 1.4. If (S, ⋆) is a binary system and C ⊆ Pr(S) is a compact convex subsystem, then there is a μ in C such that μ ⋆ μ = μ.

The following result provides an intriguing strategy for proving F ’s amenability.

Proposition 1.5. If μ ∈ Pr(𝕋) is idempotent, then μ is F -invariant.

The paper is organized as follows. Section 2 contains a review of the notation and background material which will be needed for the rest of the paper. A proof of Theorem 1.2 will be given in Section 3. This will serve as a warm-up for the more involved proof that Conjecture 1.4 implies Conjecture 1.3 in Section 4. Section 3 will also contain a proof that idempotent measures in Pr(𝕋) are F -invariant. The remaining sections contain an analysis of idempotent measures and compact convex subsystems of Pr(𝕋). These results fit roughly into two categories: those which offer some evidence which makes Conjecture 1.4 plausible and those which reveal what sort of difficulties need to be addressed in proving Conjecture 1.4.

2. Preliminaries

Before beginning, let us fix some notational conventions. In this paper, ℕ will be taken to be the positive natural numbers and ω will denote ℕ ∪ {0}. Elements of ω are identified with the set of their predecessors: 0 = ∅ and n = {0, …, n − 1}. If S is a set, then the powerset of S will be denoted by 𝒫(S).

Next we will recall the definition of a product operation on finitely additive probability measures which is an extension of the Fubini product of filters. We will need some standard definitions from functional analysis; further reading and background can be found in [21] [22]. If X is a Banach space, let X^{∗} denote the collection of continuous linear functionals on X. If S is a set, ℓ^{∞}(S) denotes the space of bounded functions from S into ℝ with the supremum norm. The space ℓ^{∞}(S)^{∗} will primarily be given the weak* topology: the weakest topology which makes the evaluation maps f ↦ f(g) continuous for each g in ℓ^{∞}(S). We will identify the collection Pr(S) of all finitely additive probability measures on S with the subspace of ℓ^{∞}(S)^{∗} consisting of those f such that f(g) ⩾ 0 for all g ⩾ 0 and such that f(¯1) = 1, where ¯1 is the function which is constantly 1; if μ is in Pr(S), then μ will be identified with the bounded linear functional f ↦ ∫ fdμ. Depending on the context, we will sometimes write f(μ) for μ(f). The elements of Pr(S) with finite support are dense in Pr(S) in the weak* topology and this will be used frequently without further mention.

Suppose that S_{0} and S_{1} are nonempty sets. Define ⊗ : Pr(S_{0}) × Pr(S_{1}) → Pr(S_{0} × S_{1}) by

μ ⊗ ν(f) = ∫ ∫ f(x, y)dν(y)dμ(x),

where f is in ℓ^{∞}(S_{0} × S_{1}). It should be noted that the order of integration is significant when measures are required to measure all subsets of S_{0} × S_{1}. This will be discussed further in Section 5.

Proposition 2.1. If S_{0} and S_{1} are nonempty sets, then for every ν ∈ Pr(S_{1}), μ ↦ μ ⊗ ν is continuous. Moreover if μ ∈ Pr(S_{0}) is finitely supported, then the map ν ↦ μ ⊗ ν is continuous.

Proposition 2.2. If S_{0}, S_{1}, and S_{2} are nonempty sets and μ_{i} ∈ Pr(S_{i}) for i < 3, then (μ_{0} ⊗ μ_{1}) ⊗ μ_{2} = μ_{0} ⊗ (μ_{1} ⊗ μ_{2}) up to the identification of (S_{0} × S_{1}) × S_{2} with S_{0} × (S_{1} × S_{2}).

Now suppose that (S, ⋆) is a binary system. Extend ⋆ to Pr(S) as follows:

μ ⋆ ν(f) = μ ⊗ ν(f ∘ ⋆) = μ ⊗ ν((x, y) ↦ f(x ⋆ y)).

It follows from Proposition 2.1 that if ν ∈ Pr(S), then μ ↦ μ ⋆ ν is continuous. If μ is finitely supported, then moreover ν ↦ μ ⋆ ν is continuous.

Thompson’s group F can be described as follows. If s, t ∈ 𝕋 have equal cardinality, then the increasing function from s to t extends linearly to an automorphism of ([0, 1], ⩽). We will write (s → t) to denote this map. The collection of all such functions with the operation of composition is F. The group F acts partially on 𝕋 by set-wise application with the stipulation that f · t is only defined when f is linear on each interval contained in the complement of t. The standard generators for F are given by:

x_{0} = ((1 1) 1 → 1 (1 1))
x_{1} = (1 ((1 1) 1) → 1 (1 (1 1)))

If we view elements of 𝕋 as terms, then the partial action of F on 𝕋 is by reassociation:

x_{0} · ((a b) c) = a (b c)
x_{1} · (s ((a b) c)) = s (a (b c))

The partial action of F on 𝕋 essentially corresponds to the action of F on its positive elements with respect to the generating set x_{k} (k ∈ ω), where x_{k+1} = x_{0}^{k}x_{1}x_{0}^{−k}. It is well known that F is amenable if and only if there is a μ in Pr(𝕋) such that

μ({t ∈ 𝕋 : x_{0} · t and x_{1} · t are defined}) = 1,
μ(x_{0} · E) = μ(x_{1} · E) = μ(E)

whenever E ⊆ 𝕋 (details can be found in, e.g., [16]). A general introduction to F and Thompson’s other groups can be found in [3].

3. A partition theorem concerning sums of at most two elements of 𝔸

In this section we will prove Theorem 1.2. The first step is to define an appropriate limit of the sequence 𝔸_{m} (m ∈ ℕ). Let U be a fixed idempotent in (βℕ, +). Define 𝔸_{U} to be the set of all μ ∈ Pr(𝕋) such that if W is open about μ, then

{m ∈ ℕ : W ∩ 𝔸_{m} ≠ ∅} ∈ U.

Define 𝕋_{U} = 𝔸_{U} ∩β𝕋. Notice that 𝔸_{U} is the U-limit of the sequence 𝔸_{m} (m ∈ ℕ) in the space K(Pr(𝕋)) consisting of all nonempty compact subsets of Pr(𝕋) equipped with the Vietoris topology. In particular, 𝔸_{U} is compact, convex, and nonempty.

Lemma 3.1. If μ and ν are in 𝔸_{U}, so is μ ν.

Proof. Let W be open about μ ν for μ and ν in 𝔸_{U}. Let Z = {p ∈ ℕ : W ∩𝔸_{p} ≠ ∅}. Since U is an idempotent, we need to prove that there is a set X in U such that for every m in X,

{n ∈ ℕ : m + n ∈ Z} ∈ U

Applying Proposition 2.1, there is an open U about μ such that if μ′ is in U, then μ′ ν is in W. By assumption, there is an X in U such that if m is in X, then U ∩ 𝔸_{m} ≠ ∅. For each m in X, let μ_{m} be an element of U ∩ 𝔸_{m}. Again by Proposition 2.1, there is, for each m in X, an open V_{m} about ν such that if ν′ is in V_{m}, then μ_{m} ν′ is in W. By our assumption that ν is in 𝔸_{U}, we have that for each m in X

Y_{m} = {n ∈ ℕ : V_{m} ∩ 𝔸_{n} ≠ ∅}

is in U. If n is in Y_{m}, fix an element ν′ of 𝔸_{n} ∩ V_{m}. It follows that μ_{m} ν′ is in W ∩ 𝔸_{m+n}. Thus Y_{m} ⊆ {n ∈ ℕ : m + n ∈ Z} and hence {n ∈ ℕ : m + n ∈ Z} is also in U. □

Theorem 1.2 will be derived from the following proposition which is of independent interest.

Proposition 3.2. If B is a finite collection of subsets of 𝕋 and U ∈ βℕ is idempotent, than there is a μ in 𝔸_{U} such that μ μ ↾ B = μ ↾ B.

Proof. By enlarging B if necessary, we may assume that it is a finite Boolean subalgebra of 𝒫(𝕋). Let A consist of the atoms A of B such that

{m ∈ ℕ : A ∩ 𝕋_{m} ≠ ∅}

is in U. Notice that if A is an atom of B which is not in A, then ξ(A) = 0 whenever ξ is in 𝔸_{U}. On the other hand, if A is in A, then there is a ξ in 𝕋_{U} such that ξ(A) = 1. Let X ⊆ 𝕋_{U} be a set of cardinality |A| such that for each A in A, there is a unique ξ in X such that ξ(A) = 1. Define a binary operation ⋆ : X × X → X by ξ ⋆ η = ζ if ζ(A) = 1 where A is the atom of B such that ξ η(A) = 1. Notice that we have ξ ⋆ η ↾ B = ξ η ↾ B. Extend ⋆ to a bilinear operation on the vector space generated by X and let C denote the convex hull of X, noting that ⋆ maps C × C into C. Notice that μ ⋆ ν ↾ B = μ ν ↾ B. Since X is finite, μ ↦ μ ⋆ μ is a continuous map defined on C. Thus there is a μ in C such that μ ⋆ μ = μ. It follows that μ μ ↾ B = μ ⋆ μ ↾ B = μ ↾ B. □

We are now ready to complete the proof of Theorem 1.2. Let c : 𝕋 → [0, 1] and ϵ > 0 be given. Fix a δ > 0 and a finite B ⊆ 𝒫(𝕋) such that if ξ, η ∈ Pr(𝕋) and |ξ(E) − η(E)| < δ for all E in B, then |ξ(c) − η(c)| < ϵ. If ξ is in ℓ^{∞}(𝕋)^{∗}, let ||ξ||_{B} denote max_{E∈B} |ξ(E)|. Fix an element μ of 𝔸_{U} such that μ ↾ B = μ μ ↾ B and set r = c(μ), recalling the convention that c(μ) = μ(c). Construct an increasing sequence μ_{i} (i ∈ ℕ) of elements of 𝔸 by induction so that if i < j, then ||μ(E) − μ_{i}(E)||_{B} < δ and

||μ μ − μ_{i} μ||_{B} < δ/2 ||μ_{i} μ − μ_{i} μ_{j}||_{B} < δ/2.

This is possible by Proposition 2.1 and the definition of 𝔸_{U}. It follows that if i ∈ ℕ, then ||μ_{i} − μ||_{B} < δ and hence |c(μ_{i}) − r| < ϵ. Simiarly if i < j, then

||μ_{i} μ_{j} − μ||_{B} ⩽ ||μ − μ_{i} μ||_{B} + ||μ_{i} μ − μ_{i} μ_{j}||_{B} < δ

and consequently |c(μ_{i} μ_{j}) − r| < ϵ. This finishes the proof of Theorem 1.2.

Extending this result even to right (or left) associated sums of at most three elements of 𝔸 seems to require new ideas. In particular, it is not clear how to find, for a given c ∈ ℓ^{∞}(𝕋), a μ in 𝔸_{U} such that c(μ) = c(μ μ) = c(μ (μ μ)).

I will conclude this section with the following proposition which relates the nonassociative form of Ellis’s Lemma to the amenability problem for F.

Proposition 3.3. If μ ∈ Pr(𝕋) is an idempotent measure, then μ is F -invariant.

Proof. Suppose that μ ∈ Pr(𝕋) satisfies μ μ = μ; we need to show that μ is F -invariant. First observe that

μ({1}) = μ μ({1}) = μ ⊗ μ(∅) = 0.

Also

μ μ({t ∈ 𝕋 : ∃a(t = a 1)}) = μ(𝕋) · μ({1}) = 1 · 0 = 0.

Since μ is an idempotent, following identities hold:

μ = μ (μ μ) = (μ μ) μ
μ = μ (μ (μ μ)) = μ ((μ μ) μ)

Now suppose that E ⊆ 𝕋.

μ(E) = μ({t ∈ E : ∃a∃b∃c(t = (a b) c)})
= (μ μ) μ({t ∈ E : ∃a∃b∃c(t = (a b) c)})
= (μ ⊗ μ) ⊗ μ({(a, b, c) ∈ 𝕋^{3} : (a b) c ∈ E})
= μ ⊗ (μ ⊗ μ)({(a, b, c) ∈ 𝕋^{3} : a (b c) ∈ x_{0} · E})
= μ (μ μ)({t ∈ x_{0} · E : ∃a∃b∃c(t = a (b c))}) = μ(x_{0} · E).

A similar computation shows that μ(x_{1} · E) = μ(E). □

4. Conjecture 1.4 implies Conjecture 1.3

In this section I will prove that the nonassociative form of Ellis’s Lemma implies the nonassociative form of Hindman’s Theorem. This will be an elaboration on the proof of Theorem 1.2 from Proposition 3.2.

Before proceeding, it is necessary to define the notion of admissibility from the statement of Conjecture 1.3. In order to motivate the definition of admissibility, consider the natural adaptation of the proof in the previous section: at stage n, we have constructed measures μ_{i} (i < n) and wish to pick a measure μ_{n} in 𝔸 such that if t is any element of 𝕋 and i_{k} (k < m) is an increasing sequence of length at most #(t), then c(t(μ_{i_{0}}, …, μ_{i_{m−1}}, μ, μ, …, μ)) and c(t(μ_{i_{0}}, …, μ_{i_{m−1}}, μ_{n}, μ, …, μ) differ by less than ϵ2^{−n−1}. The problem is that there are infinitely many t’s to consider. The fix to this problem is to consider only those t of cardinality at most n at stage n. This readily allows us to prove the form of Conjecture 1.3 in which one obtains the conclusion for those sequences i_{k} (k < m) satisfying m − 1 ⩽ i_{0}.

It is possible to do better, however, by noticing that if μ is idempotent, then expressions such as t(μ_{i_{0}}, …, μ_{i_{m−1}}, μ, μ, …, μ) may allow for some algebraic simplification. This is made precise as follows. For each k < #(t), define t ↦ t_{k} by

t_{k} = \begin{cases}1 & if t = 1 and k = 0 \\ a_{k} 1 & if t = a b and k < #(a) \\ a b_{k−#(a)} & if t = a b and #(a) ⩽ k < #(t) \end{cases}

Notice that if we regard t as a rooted orderd binary tree, then t_{k} is the result of iteratively removing all carets in t which involve only leaves of index greater than k (where leaves are indexed in increasing order from left to right). Set l_{k}(t) = #(t_{k}) − 2. If t is in 𝕋_{m}, then an increasing sequence i_{k} (k < m) is admissible for t if for all k < m, l_{k}(t) ⩽ i_{k}.

Notice that l_{k}(t) < #(t) for each k and in particular a sequence i_{k} (k < m) is admissible for any element of 𝕋_{m} provided that m ⩽ i_{0}. Also, the value of l_{k} at any right associated power of 1 is at most k and hence any increasing sequence is admissible for some element of 𝕋_{m}.

It will be helpful to adopt the following notation: if t is in 𝕋_{m}, k ⩽ m, and ν_{i} (i < k) and μ are in Pr(𝕋), let t(ν_{0}, …, ν_{k−1}; μ) denote t(ν_{0}, …, ν_{k−1}, μ, …, μ) (i.e. the sequence ν_{i} (i < k) is extended to a sequence of length m by adding on a sequence of m − k many μ’s and then substituting into t). The key property of the definition of t_{k} is that whenever t in 𝕋, ν_{i} (i < k) are in 𝔸, and μ is an idempotent in Pr(𝕋), then t(ν_{0}, …, ν_{k−1}; μ) = t_{k}(ν_{0}, …, ν_{k−1}; μ); this is easily established by induction on #(t).

We are now ready to prove the main result of the section.

Theorem 4.1. Conjecture 1.4 implies Conjecture 1.3.

Proof. In Section 3, we proved that if U is an idempotent in (βℕ, +), then 𝔸_{U} is a nonempty compact convex subsystem of (Pr(𝕋), ). Thus if Conjecture 1.4 is true, then 𝔸_{U} contains a μ such that μ μ = μ. Fix a c : 𝕋 → [0, 1] and set r = c(μ). Construct an increasing sequence μ_{i} (i ∈ ω) in 𝔸 by recursion such that, if μ_{i} (i < n) have been constructed, then for all k < m ⩽ n + 2 and i_{0} < … < i_{k−1} < n and t in 𝕋_{m},

|c(t(μ_{i_{0}}, …, μ_{i_{k−1}}; μ)) − c(t(μ_{i_{0}}, …, μ_{i_{k−1}}, μ_{n}; μ))| < ϵ2^{−n−1}.

This is possible by applying the definition of 𝔸_{U} and the following claim.

Claim 4.2. If t is in 𝕋_{m} and ν_{i} (i < m) are such that ν_{i} is in Pr(𝕋) and has finite support if i < k, then the function F defined by

F (ζ) = t(ν_{0}, …, ν_{k−2}, ζ, ν_{k}, …, ν_{m−1})

is continuous.

Proof. The proof is by induction on m. If m = 1, then there is nothing to show since then F is just the identity. If m > 1 and t is in 𝕋_{m}, then there are a and b such that t = a b. If #(a) = l ⩽ k, then

t(ν_{0}, …, ν_{k−2}, ζ, ν_{k}, …) = a(ν_{0}, …, ν_{l−1}) b(ν_{l}, …, ν_{k−1}, ζ, ν_{k}, …)

Letting ν = a(ν_{0}, …, ν_{l−1}), we have that

F (ζ) = ν b(ν_{l}, …, ν_{k−1}, ζ, ν_{k}, …, ν_{m−1}).

Continuity of F now follows from Proposition 2.1 and the induction hypothesis applied to b. A similar argument handles the case #(a) > k. □

Now we will verify that μ_{k} (k ∈ ω) satisfies the conclusion of Conjecture 1.3. To this end, let t be an element of 𝕋_{m} and let i_{0} < … < i_{m−1} be admissible for t. By construction we have

|c(t(μ_{i_{0}}, …, μ_{i_{m−1}})) − c(t(μ, …, μ))| ⩽

∑_{k<m} |c(t(μ_{i_{0}}, …, μ_{i_{k}}; μ)) − c(t(μ_{i_{0}}, …, μ_{i_{k−1}}; μ))|

= ∑_{k<m} |c(t_{k}(μ_{i_{0}}, …, μ_{i_{k}}; μ)) − c(t_{k}(μ_{i_{0}}, …, μ_{i_{k−1}}; μ))|.

By admissibility, #(t_{k}) = l_{k}(t) + 2 ⩽ i_{k} + 2 and thus μ_{i_{k}} was chosen such that

|c(t_{k}(μ_{i_{0}}, …, μ_{i_{k}}; μ)) − c(t_{k}(μ_{i_{0}}, …, μ_{i_{k−1}}; μ))| < ϵ2^{−i_{k}−1}.

Recalling that r = c(t(μ, …, μ)) and putting this all together we have that

|c(t(μ_{i_{0}}, …, μ_{i_{m−1}})) − r| < ϵ ∑_{k} 2^{−k−1} = ϵ.

□

We finish this section by recalling a result of [17] which asserts that a weak form of the finitary version of Conjecture 1.3 is in fact equivalent to the amenability of F. If m ∈ ℕ, an embedding of 𝕋_{m} in 𝕋 is function of the form

t ↦ t(u_{0}, …, u_{m−1})

for some sequence u_{i} (i < m) of elements of 𝕋. Observe that such an embedding maps into 𝕋_{n} for some n. An embedding of 𝕋_{m} into 𝔸_{n} is a convex combination of embeddings of 𝕋_{m} into 𝕋_{n}. The range of such an embedding is a copy of 𝕋_{m} in 𝔸_{n}.

Theorem 4.3. [17] The following are equivalent:

(1) Thompson’s group F is amenable.

(2) For every m there is an n such that if c : 𝕋_{n} → [0, 1], then there is a copy 𝕏 of 𝕋_{m} in 𝔸_{n} such that |c(ν) − c(ν′)| ⩽ 1/2 whenever ν, ν′ ∈ 𝕏.

(3) For every m there is an n such that if c : 𝕋_{n} → {0, 1}, then there is a copy 𝕏 of 𝕋_{m} in 𝔸_{n} such that c is constant on 𝕏.

Remark 4.4. A more restrictive notion of copy is the following: a strong copy of 𝕋_{m} in 𝔸 is the range of a function of the form

t ↦ t(μ_{0}, …, μ_{m−1})

where μ_{i} (i < m) are elements of 𝔸. The finite form of Conjecture 1.3 would assert that for every m there is an n such that if f : 𝕋_{n} → [0, 1], then there is a strong copy of 𝕋_{m} in 𝔸_{n} on which f is within 1/2 of being constant. It is unclear if this assertion is equivalent to the amenability of F.

5. Monotonicity properties of F-invariant measures

In [16], a lower bound was established for the Følner function for F, assuming that it is amenable. This was achieved by establishing and analyzing the following qualitative property of F -invariant measures in Pr(𝕋), which is of independent interest.

Proposition 5.1. [16] If μ is an F -invariant measure in Pr(𝕋), then for μ-a.e. t ∈ 𝕋, one of the following pairs of inequalities holds:

#(t/001) < #(t/01) < #(t/10)
#(t/10) < #(t/01) < #(t/001).

Here t/σ is defined recursively as follows: t/σ = t if σ is the empty string and

(a b)/0σ = a/σ (a b)/1σ = b/σ.

That is, t/σ is the sub-term of t located at address σ. The proof of the above proposition generalizes to yield the following result. Recall that a quasi-order is a reflexive, transitive relation; a quasi-order ⪯ is linear if for every x and y, either x ⪯ y or y ⪯ x.

Proposition 5.2. Suppose that ⪯ is a linear quasi-order on 𝕋 such that if s, t ∈ 𝕋, then s, t ≺ s t. If μ ∈ Pr(𝕋) is an F -invariant measure, then either

• for every incompatible pair σ <_{lex} ς of nonconstant finite binary sequences, μ-a.e. t satisfy t/σ ≺ t/ς or

• for every incompatible pair σ <_{lex} ς of nonconstant finite binary sequences, μ-a.e. t satisfy t/ς ≺ t/σ.

Remark 5.3. It is worth noting an example of a linear quasi-order on 𝕋 which is quite different than the order defined by #(s) ⩽ #(t). Define an equivalence relation ≡_{LD} on 𝕋 by relating two elements if they can be proved equal using the left self distributive law a (b c) = (a b) (a c). Define a <_{LD} b if there are a′ and b′ which are ≡_{LD}-equivalent to a and b, respectively, such that a′ is a subterm of b′ (i.e. a′ = b′/σ for some σ of positive length). By work of Laver and Dehornoy [13] [4], if a and b are in 𝕋, then exactly one of the following is true: a <_{LD} b, a ≡_{LD} b, or b <_{LD} a. Set a ⩽_{LD} b if a ≡_{LD} b or a <_{LD} b. It follows that ⩽_{LD} a linear quasi-order which satisfies the hypothesis of Proposition 5.2. Notice that except for the left associated elements of 𝕋, which do not allow any application of the left self distributive law, the equivalence classes of ≡_{LD} are infinite.

The purpose of this section is to prove the following proposition, which shows that a natural strategy for proving Conjecture 1.4 does at least yield measures with this qualitative property.

Proposition 5.4. Suppose C ⊆ Pr(𝕋) is minimal with respect to the properties of being compact, convex, -closed, and nonempty and let ⪯ be a linear quasi-order on 𝕋 such that for all s, t ∈ 𝕋, s, t ≺ s t. Either

• for every μ ∈ C and incompatible pair σ <_{lex} ς of finite binary sequences, μ-a.e. t satisfy t/σ ≺ t/ς or

• for every μ ∈ C and incompatible pair σ <_{lex} ς of finite binary sequences, μ-a.e. t satisfy t/ς ≺ t/σ.

Proof. Let C be given as in the statement of the proposition. First observe that the convex hull of C C is dense in C. This is because the closure of the convex hull of C C is clearly nonempty, compact, convex, and -closed (in fact if C C ⊆ X ⊆ C, then X is -closed). We will need the following claim.

Claim 5.5. If I ⊆ 𝕋 is an interval in (𝕋, ⪯), then either μ(I) = 1 for every μ in C or μ(I) = 0 for every μ in C.

Proof. First observe that it suffices to prove the claim if I is an initial segment of 𝕋 in the ⪯-order. Suppose that there exists a μ in C such that μ(I) = p < 1. Let C′ = {ν ∈ C : ν(I) ⩽ p^{2}}. Observe that C′ is compact and convex. Next notice that if s and t are in 𝕋 and s t is in I, then both s and t are in I. Thus

μ μ(I) = μ μ({s t ∈ 𝕋 : s t ∈ I})
⩽ μ μ({s t ∈ 𝕋 : s, t ∈ I}) = μ μ(I I) = μ(I) · μ(I) = p^{2}

and hence μ μ is in C′. Furthermore, if ξ and η are in C′, then a similar computation shows that ξ η(I) ⩽ p^{4} ⩽ p^{2} and thus that C′ is -closed. We have therefore showed that C′ ⊆ C is compact, nonempty, and -closed. It follows that C′ = C and thus that μ is in C′. But this means p ⩽ p^{2}. Since p < 1 holds by assumption, we have that p = 0. It follows that every element of C = C′ assigns measure 0 to I. □

Let I denote the set of all s in 𝕋 such that for every μ in C,

μ({t ∈ 𝕋 : s ≺ t}) = 1

and observe that I is an initial part of 𝕋. Define J = 𝕋 ∖ I.

The remainder of the proof breaks into two cases. First consider the case that μ(I) = 1 for every μ in C. We will need the following claim.

Claim 5.6. For each μ in C, each s in I, and each finite binary sequence σ

μ({t ∈ 𝕋 : t/σ ∈ I and s ≺ t/σ}) = 1.

Proof. Fix s ∈ I; the proof is by induction on the length of σ. We have already established the base case. In order to handle the inductive step, let σ be a given finite binary sequence of positive length. Notice that the set A of μ in C such that

μ({t ∈ 𝕋 : t/σ ∈ I and s ≺ t/σ}) = 1

is compact and convex. Thus it suffices to show that C C ⊆ A. Observe that if σ = 0σ̄, then (u v)/σ = u/σ̄ and if σ = 1σ̄, then (u v)/σ = v/σ̄. If ξ η is in C C and σ = 0σ̄, then the claim follows from

ξ η({t ∈ 𝕋 : t/σ ∈ I and s ≺ t/σ}) = ξ({u ∈ 𝕋 : u/σ̄ ∈ I and s ≺ u/σ̄})

which equals 1 by the induction hypothesis. The case σ = 1σ̄ is analogous. □

We will now prove the first alternative of the proposition by induction on the maximum length of the sequences. As in the claim, the set of μ which satisfy the first alternative of the proposition is compact and convex and hence it is sufficient to show that it contains C C. Let σ <_{lex} ς be given finite binary sequences. If σ and ς both begin with the same digit, then observe that if ξ η is in C C, then the induction hypothesis implies that for ξ η-a.e. t, t/σ ≺ t/ς. If σ = 0σ̄ and ς = 1ς̄, then fix a u such that u/σ̄ is in I. If ξ η is in C C, then for η-a.e. v, u/σ̄ ≺ v/ς̄. Notice that if t = u v, then t/σ = u/σ̄ and t/ς = v/ς̄. Since ξ-a.e. u satisfies that u/σ̄ is in I, it follows that for ξ η-a.e. t, t/σ ≺ t/ς. This finishes the proof of the case μ(I) = 1 for all μ in C.

Now suppose that μ(J) = 1 for all μ in C. This case is almost identical to the proof of the previous case, but with the observation that, by definition of I and Claim 5.5, we have that μ({t ∈ 𝕋 : s ⪯ t}) = 0 whenever s is in J and μ is in C. □

6. The set of idempotents in 𝔸_{U}

The purpose of this section is to show that if 𝔸_{U} contains any idempotents, it contains at least 2^{ℵ_{0}} of them. This is then used to show that if there any idempotents in 𝔸_{U}, then the set of idempotents in 𝔸_{U} is far from being closed.

Proposition 6.1. Suppose that {μ_{i} : i ∈ ω} consists of idempotents and that there are sets E_{i} ⊆ 𝕋 such that μ_{i}(E_{j}) = 1 if i = j and 0 otherwise. Any limit point of {μ_{i} : i ∈ ω} is not an idempotent.

Proof. Suppose that μ_{i} (i ∈ ω) and E_{i} (i ∈ ω) are as in the statement of the proposition and let μ be a limit point of {μ_{i} : i ∈ ω}. Let t_{i} (i ∈ ω) list 𝕋 and define

E = ⋃_{j<i} t_{i} E_{j}.

Observe that for each i, μ(⋃_{j<i} E_{j}) = 0 and therefore μ μ(E) = 0. On the other hand, for each k ∈ ω, μ_{k}(⋃_{j<i} E_{j}) = 1 whenever k < i. Since idempotent measures assign measure 0 to finite sets, it follows that

μ_{k}(E) = μ_{k} μ_{k}(E) = ∫ ∫ χ_{E}(s t) dμ_{k}(t) dμ_{k}(s) = 1.

Since μ is a limit point of {μ_{k} : k ∈ ω}, it follows that μ(E) = 1 ≠ μ μ(E) = 0. □

Proposition 6.2. If 𝔸_{U} contains an idempotent, then there exist idempotents {μ_{r} : r ∈ 2^{ω}} ⊆ 𝔸_{U} and sets E_{r} ⊆ 𝕋 for r ∈ 2^{ω} such that μ_{r}(E_{s}) = 1 if r = s and 0 otherwise. In particular, the set of idempotents in 𝔸_{U} is either empty or else is not a closed set.

Proof. Begin by defining u_{σ} by induction on the length of σ, whenever σ a nonempty finite binary sequence. If σ has length 1, define u_{σ} = 1. If u_{σ} has been defined, set

u_{0σ} = u_{σ} 1 u_{1σ} = 1 u_{σ}

and observe that #(u_{σ}) coincides with the length of σ. For r in 2^{ω} and n ∈ ω, define E_{r,n} to be the subsystem of (𝕋, ) generated by {u_{r↾k} : n < k} and let E_{r} denote E_{r,0}. Observe that if r ≠ s, then for sufficiently large n, E_{r,n} is disjoint from E_{s}.

For r ∈ 2^{ω}, define h_{r} : 𝕋 → E_{r} by

h_{r}(t) = \begin{cases}h_{r}(a) h_{r}(b) & if t = a b and a << b \\ u_{r↾#(t)} & if t is not as above. \end{cases}

Here a << b means that for some p, #(a) < 2^{p} and 2^{p} divides #(b). Extend h_{r} to a continuous linear operator on ℓ^{∞}(𝕋), also denoted by h_{r} as follows:

h_{r}(μ)(f) = μ(f ∘ h_{r}) = ∫ f(h_{r}(t)) dμ(t).

Notice that h_{r} maps each 𝔸_{m} into 𝔸_{m} and hence maps 𝔸_{U} into 𝔸_{U}.

The proposition follows from the next two claims. First observe that if p is in ℕ and μ is in 𝔸_{U}, then

μ({t ∈ 𝕋 : 2^{p} divides #(t)}) = 1

This follows from the fact that ℤ/2^{p}ℤ has a unique idempotent and that the canonical homomorphism from (ℕ, +) into ℤ/2^{p}ℤ extends to a homomorphism of (βℕ, +) into ℤ/2^{p}ℤ which must send U to 0.

Claim 6.3. If μ and ν are in 𝔸_{U}, then h_{r}(μ ν) = h_{r}(μ) h_{r}(ν).

Proof. Let μ and ν be fixed. Notice that, for a fixed s, ν-a.e. t satisfies that s << t and thus that h_{r}(s t) = h_{r}(s) h_{r}(t). The claim now follows by unfolding the definitions:

h_{r}(μ ν)(f) = ∫ ∫ f(x) dh_{r}(μ ν)(x) = ∫ ∫ f(h_{r}(y)) dμ ν(y)

= ∫ ∫ f(h_{r}(s t)) dν(t) dμ(s) = ∫ ∫ f(h_{r}(s) h_{r}(t)) dν(t) dμ(s)
= ∫ ∫ f(u v) dh_{r}(ν)(v) dh_{r}(μ)(u) = h_{r}(μ) h_{r}(ν)(f)
□

Claim 6.4. If μ is in 𝔸_{U}, then h_{r}(μ)(E_{r,n}) = 1 for all n and if s ≠ r then μ(E_{s}) = 0.

Proof. Observe that if 2^{p} divides a + b and a << b, then 2^{p} divides both a and b. Consequently, if 2^{p} divides #(t), then h_{r}(t) is in E_{r,2^{p}}. Now suppose that μ in 𝔸_{U}. Since, for every p, μ-a.e. t satisfies 2^{p} divides #(t), it follows that h_{r}(μ)(E_{r,2^{p}}) = 1. The second conclusion of the claim follows from the observation that if r ≠ s, then there is an n such that E_{r,n} ∩ E_{s} is empty. □

To finish the proof of the proposition, let μ be any element of 𝔸_{U} such that μ μ = μ. Define μ_{r} = h_{r}(μ). The above claims show that μ_{r} μ_{r} = μ_{r} and that μ_{r}(E_{s}) = 1 if r = s and 0 otherwise. This finishes the proof of the proposition. □

In September 2012, I publicly announced a claim that Thompson’s group F is amenable. The proof contained an error, found by Azer Akhmedov. The portion of the proof that was incorrect hinged on a claim that the set of idempotent measures in 𝔸_{U} was equal to a directed intersection of nonempty compact sets. In light of the above propositions, this is impossible.

Of course the proofs in this section also suggest why this is the wrong approach: 𝔸_{U}, which is the closed convex hull of β𝕋 ∩ 𝔸_{U}, is too large. At present, it seems plausible that if K is a minimal compact subsystem of (β𝕋, ), then the closed convex hull of K contains a unique idempotent. Next we will show, however, that such K are necessarily quite large.

Proposition 6.5. If K is a minimal compact subsystem of (β𝕋, ), then K is nonseparable.

Proof. If x is in 2^{ℕ}, define x + 1 to be the result of adding 1 to x with carry to the right (i.e. x ↦ x + 1 is the odometer map). Define h : 𝕋 → 2^{ℕ} by setting h(1) to be the constant 0 sequence and h(s t) = h(t) + 1 and extend h continuously to a map from β𝕋 to 2^{ℕ}. Notice that the h-image of any nonempty -closed subset of β𝕋 is dense and hence the image of K is all of 2^{ℕ} (since the odometer is a minimal dynamical system). Moreover, for each r in 2^{ℕ}, there is a ξ_{r} in K such that h(ξ_{r} ξ_{r}) = r. For each r in 2^{ℕ} and p in ℕ, define

E_{r,p} = {t ∈ 𝕋 : ∀i < p (h(t)(i) = r(i))}.

Notice that ξ_{r}(E_{r−1,p}) = 1 for all p. Define

E_{r} = {s t : s t ∈ E_{r,#(s)}} = {s t : t ∈ E_{r−1,#(s)}}

and observe that ξ_{r} ξ_{r}(E_{r}) = 1. As in the previous section, the minimality of K implies that K K is dense in K. It follows that if

U_{r} = {η ∈ K : η(E_{r}) > 1/2},

then {U_{r} : r ∈ 2^{ℕ}} is an uncountable family of nonempty pairwise disjoint open subsets of K. Hence K is not separable. □

7. Concluding remarks

At the time this article was written, it is still unknown if F is amenable. Neverthe-less, I feel Conjectures 1.3 and 1.4 are based on sound heuristics from Ramsey theory. It is rare in the Ramsey theory of countably infinite sets that there are difficult counterexamples to Ramsey-theoretic statements (there are exceptions, perhaps most notably [18]; see also [23, §9]). On the other hand, there are many deep and often difficult positive results in Ramsey theory at this level: the Dual Ramsey Theorem [8], Hindman’s Theorem [11], Gowers’s FIN_{k} Theorem [7], the Hales-Jewett Theorem [9], and the Halpern-Läuchli Theorem [10]. See [25] for further reading on these theorems as well as many others.

Also, while we do not know whether (Pr(S), ⋆) contains an idempotent if (S, ⋆) is an arbitrary binary system, we do know that there are quite different examples of binary systems which admit idempotent measures: semigroups, finite binary systems, and binary systems depending on only one variable.

The results of this paper also suggest several test questions which allow for an incremental approach to proving Conjectures 1.3 and 1.4:

Question 7.1. Is Conjecture 1.3 true for sums of d elements, for a fixed d ⩾ 3? What about the case d = 3?

Question 7.2. For which classes of binary systems is Conjecture 1.4 true?

Question 7.3. If two binary systems satisfy Conjecture 1.4, does their product?

Question 7.4. For which specific values of m can one prove that there is an n such that if c : 𝕋_{n} → {0, 1} then there is a copy of 𝕋_{m} in 𝔸_{n} on which c is constant? What bounds (upper or lower) can be proved on n for a given value of m?

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Justin Moore, 555 Malott Hall, Department of Mathematics, Cornell University, Ithaca, NY 14853-4201