Amenability problem for Thompson's group $F$: state of the art

This is a survey of our recent results on the amenability problem for Thompson's group $F$. They mostly concern esimating the density of finite subgraphs in Cayley graphs of $F$ for various systems of generators, and also equations in the group ring of $F$. We also discuss possible approaches to solve the problem in both directions.


Preliminaries
In this survey we collect a number of our recent results on the amenability problem for Thompson's group F .They are mostly contained in [27,28,29,30,31,32,33,34]. In the this section we introduce concepts and notation used throughout the paper.The reader who is familiar with that can skip this background.
We recall the concept of a graph in the sense of Serre, Cayley graph of a group (right anf left), internal and external vertices of a subgraph of a Cayley graph, and various kinds of its boundary.It is convenient to use automata terminology saying that an automaton accepts (or does not accept) a generator.We introduce an important concept of density of Cayley graphs and their subgraphs together with the related concept of their (Cheeger) isoperimetric constant.
Then we recall the definition of amenabilty of semigroups and groups, discuss Folner criterion and some other necessary and sufficient conditions for amenabilty.In order to work with elements of Thompson's group F whose basic properties are discussed in a separate subsection, we present terminology related to rooted binary trees and forests.We recall the definition of the group F and list some of its important properties.The amenability problem for this group is in the centre of our attention.
Section 2 contains the description of our recent result on the density of finite subgraphs of the Cayley graph of F in various generating systems.One of the most important results is Theorem 2.7 where we show that the density of the Cayley graph of F in standard generators strictly exceeds 3.5.This is an improvement of the Belk-Brown construction for which there was a conjecture of its optimality.Section 3 concerns equations in the group ring of F and their systems.Due to results by Bartholdi and Kielak, we know that the amenabilty problem for F is equivalent to the Ore condition for group rings, that is, the question whether equation au = bv in this ring has nonzero solutions.We present a number of results in this direction giving a reduction of the problem to the case of monoid rings K[M ] of the positive monoid M ⊂ F and homogeneous polynomials a, b.We solve some partial cases of this problem including the case of linear polynomials.We also give a description for the set of solutions of some important equations and their systems.
Section 4 contains a new description of exhausting finite subsets of the Cayley graph of F and a new algorithm to solve the word problem in this group.Besides, we discuss possible approaches to the amenability problem to solve it in both directions.
Let A be an alphabet, that is, a nonempty set of symbols (or letters).Taking a disjoint bijective copy A −1 of the set A, we get the set A ±1 = A ∪ A −1 called a group alphabet.It has an involution −1 without fixed points defined by (a −1 ) −1 ⇌ a. Elements of the free monoid (A ±1 ) * are called group words.
Let G be a group equipped by mapping A → G such that the image of A generates G.We will say that A is a set of (group) generators for G. Usually A is identified with its image provided the above map is injective.For more general needs it is convenient to include the case when different symbols may denote the same group generator.In this case we may say that A defines a multiset of generators, say, {a 1 = x 0 , a 2 = x 0 , a 3 = x 1 }, where x 0 is doubly repeated.
Let G be a group generated by A in the above sense.Its right Cayley graph Γ r = C(G; A) is defined as follows.The set of vertices is G; for any a ∈ A ±1 we put a directed edge e = (g, a) with ι(e) = g, τ (e) = ga.Its inverse is e −1 = (ga, a −1 ).Each edge has a label; for the above edge e = (g, a) it is just a.The label function can be naturally continued to the set of paths.
The concept of a left Cayley graph Γ l = C(G; A) is defined in a similar way.Here the set of vertices is also G.A directed egde e = (a, g) with label a has ι(e) = ag, τ (e) = g.That is, going along the edge labelled by a, means cancelling a on the left.The inverse egde is e −1 = (a −1 , ag).Labels of paths are defined similarly for this case.Let G be an infinite group generated by A. For our needs we assume that A is always finite, |A| = m.Let Γ = C(G, A) be the Cayley graph of G, right or left.To any finite nonempty subset Y ⊂ G we assign a subgraph in Γ adding all edges connecting vertices of Y .So given a set Y , we will usually mean the corresponding subgraph.This is a labelled graph that we often call an automaton.For each g ∈ Y we have exactly 2m directed edges in Γ starting at g, where a ∈ A ±1 .If the endpoint of such an edge with label a belongs to Y , then we say that the vertex g of our automaton Y accepts a.For the case of right Cayley graphs this means ga ∈ Y , for the case of left Cayley graphs this means a −1 g ∈ Y .
A vertex g ∈ Y is called internal whenever it accepts all labels a ∈ A ±1 .That is, the degree of g in Y equals 2m.Otherwise we say that g belongs to the inner boundary of Y denoted by ∂Y .
By dist(u, v) we denote the distance between two vertices in Γ, that is, the length of a shortest path in Γ that connects vertices u, v.For any vertex v and a number r ≥ 0 let B r (v) denote the ball of radius r around v, that is, the set of all vertices in Γ at distance ≤ r from v. To any vertex v in ∂Y we can assign an external edge starting at v.This gives an injection from ∂Y to ∂ * Y .On the other hand, there exist at most 2m edges in By the density of a subgraph Y we mean its average vertex degree.This concept was introduced in [26]; see also [27] (We can mention also the paper [2] where some computaional experiments with densities of subgraphs are presented.) By the density of the Cayley graph Γ = C(G; A) we mean the number δ where Y runs over all nonempty finite subgraphs.Analogously, the Cheeger isoperimetric constant of the group G with respect to generating set A, is the number Clearly, δ(Γ) + ι * (Γ) = 2m.
In further sections, working with isoperimetric constants, we will write ι * (G; A) instead of the above notation.
1.3.Amenability of semigroups and groups.Let S be a semigroup.Suppose that there exists a mapping µ : P(S) → [0, 1] from the power set of S into the unit interval satisfying the following conditions: Then the semigroup S is called left amenable.
The definition of right amenable semigroups is given in a similar way.These concepts differ in general.However, for the case of groups they are equivalent.Moreover, both of them will be equivalent to the conditions with the two-sided invariance.The proof can be found in [21].
We do not list all well-known properties of (non)amenable groups.It is sufficient to refer to one of modern surveys like [42].Just notice that all finite and abelian groups are amenable.The class of amenable groups is closed under taking subgroups, homomorphic images, group extensions, and directed unions of groups.The groups in the closure of the classes of finite and abelian groups under this list of operations are called elemetary amenable (EA).Also we need to say that free groups of rank > 1 are not amenable.
We will often refer to the following Følner criterion [18].Here we restrict ourselves to the case of finitely generated groups.If one expects that a group is not amenable, or the answer is unknown, it is reasonable to construct sets with the least possible value of ι * (G; A).We will discuss this approach later.
In case when we are going to establish non-amenability of a group, it is useful to apply some other criteria.
Let G be a group generated by a finite set A. A doubling function on G is a mapping ψ : G → G such that a) for all g ∈ G the distance dist(g, ϕ(g)) is bounded from above by a constant K > 0, b) any element g ∈ G has at least two preimages under ψ.
Proposition 1.2.A group G with finite set of generators A is non-amenable if and only if it admits a doubling function.
This criterion is often attributed to Gromov.An elegant proof of it can be found in [11], see also [14].Note that this property also does not depend on the choice of a finite generating set.
An interesting partial case happens if the constant K in the above definition equals 1.In [26] we proved that for a 2-generator group G, a doubling function with constant K = 1 exists if and only if the density of the Cayley graph does not exceed 3. Equivalently, one can say that the Cheeger isoperimetric constant is at least one: ι * (G; A) ≥ 1.
We call a group G strongly non-amenable (with respect to a given finite generating set A) whenever this inequality holds: ι * (G; A) ≥ 1.It is interesting to find a nesessary and sufficient condition for this property for groups with any finite number of generators.This will be done in Section 2.
Another practical criterion for non-amenabilty can be stated in terms of flows on Cayley graphs.Let us introduce the terminology for that.
A flow on a graph Γ is a real-valued function f : E → R such that f (e −1 ) = −f (e).We say that f (e) is the flow through the edge e.Given a vertex v, we define an inflow to it as a sum of flows through all edges with v as a terminate vertex.
The following criterion in terms of flows is essentially known.It can be derived from the above criterion in terms of doubling functions.Proposition 1.3.Let G be a group with finite generating set A, and let Γ = C(G; A) be its Cayley graph.The group G is non-amenable if and only if there exist constants C > 0 and ε > 0, and a flow f on Γ with the following properties: a) The absolute value of the flow through each edge is bounded: |f (e)| ≤ C for all e ∈ E; b) The inflow into each vertex is at least ε.
1.4.Rooted binary trees and forests.We add this short subsection to introduce some notation used in the paper.Formally, a rooted binary tree can be defined by induction.1) A dot • is a rooted binary tree.
2) If T 1 , T 2 are rooted binary trees, then (T 1 ˆT2 ) is a rooted binary tree.
3) All rooted binary trees are constructed by the above rules.
Instead of formal expressions, we will use their geometric realizations.A dot will be regarded as a point.It coincides with the root of that tree.If T = (T 1 ˆT2 ), then we draw a caret forˆas a union of two closed intervals AB (goes left down) and AC (goes right down).The point A is the root of T .After that, we draw trees for T 1 , T 2 and attach their roots to B, C respectively in such a way that they have no intersection.It is standard that for any n ≥ 0, the number of rooted binary trees with n carets is equal to the nth Catalan number c n = (2n)!n!(n+1)! .Each rooted binary tree has leaves.Formally, they are defined as follows: for the one-vertex tree (which is called trivial ), the only leaf coincides with the root.In case T = (T 1 ˆT2 ), the set of leaves equals the union of the sets of leaves for T 1 and T 2 .In this case the leaves are exactly vertices of degree 1.
We will also need the concept of a height of a rooted binary tree.For the trivial tree, its height equals 0. For T = (T 1 ˆT2 ), its height is ht Now we define a rooted binary forest as a finite sequence of rooted binary trees T 1 , ... , T m , where m ≥ 1.The leaves of it are the leaves of the trees.It is standard from combinatorics that the number of rooted binary forests with n leaves also equals c n .The trees are enumerated from left to right and they are drawn in the same way.
A marked (rooted binary) forest is a (rooted binary) forest where one of the trees is distinguished.
1.5.Thompson's group F .We define the Richard Thompson group F in a combinatorial way, using the following infinite group presentation (1.1)This group was found by Richard J. Thompson in the 60s.We refer to the survey [10] for details.(See also [6,7,8].)A recent survey on the subject with respect to the amenability problem of F can be found in [30].
It is easy to see from the relations of (1.1) that for any n ≥ 2, one has x n = x −(n−1) 0 x 1 x n−1 0 so the group is generated by x 0 , x 1 .It can be given by the following presentation with two defining relations Each element of F can be uniquely represented by the normal form, that is, an expression of the form Equivalent definitions of F can be given in terms of piecewise-linear functions.Although these definitions are very popular, we do not describe it here since we will not use them in our survey.
It is known from [25] that F is the diagram group over the simplest semigroup presentation⟨ x | x = x 2 ⟩.This way to represent elements of F is very useful for many situations.Sometimes it is preferable to use non-spherical diagrams over the same presentation instead of spherical ones.The latter approach was described in [26].Diagrams can be replaced by dual graphs.This leads to a standard way to represent elements of F as pairs of rooted binary trees.For the needs of this survey, it suffices to work with elements of the positive monoid M repersenting them as marked binary forests according to Section 1.4.
The group F has no free non-abelian subgroups [6].It is known [12] that F is not elementary amenable.However, the famous problem about amenability of F remains open.The question whether F is amenable was asked by Ross Geoghegan in 1979; see [19,20].There is no common opinion on the answer: some of specialists in this area try to prove non-amenability, some of them believe that the group is amenable.There is a number of papers with attempts to solve the problem in both directions.The author always believed in non-amenability of F trying to prove this property.Now we are not sure on that answer, the reasons will be explained later.
If F is amenable, then it is an example of a finitely presented amenable group, which is not EA.If it is not amenable, then this gives an example of a finitely presented group, which is not amenable and has no free non-abelian subgroups.Note that the first example of a nonamenable group without free non-abelian subgroups has been constructed by Ol'shanskii [40].The question about such groups was formulated in [13], it is also often attributed to von Neumann [39].Adian [1] proved that free Burnside groups with m > 1 generators of odd exponent n ≥ 665 are not amenable.The first example of a finitely presented non-amenable group without free non-abelian subgroups has been constructed by Ol'shanskii and Sapir [41].Grigorchuk [24] constructed the first example of a finitely presented amenable group not in EA.
It is not hard to see that F has an automorphism given by To check that, one needs to show that both defining relators of F in (1.2) map to the identity.This is an easy calculation using normal forms.After that, we have an endomorphism of F .Aplying it once more, we have the identity map.So this is an (outer) automorphism of order 2.
Later we will add more arguments to the importance of the symmetric set S = {x 1 , x1 }, where x1 = x 1 x −1 0 .Obviously, S also generates F .We let Applying Tietze transormations to (1.2) one can get a presentation of F in the new generating set: (1.4) From the symmetry reasons we know that β α ↔ α β 2 also holds in F .Therefore, it is a consequence of the two relations of (1.4).Moreover, one can check that for any positive integers m, n it holds α β m ↔ β α n as a consequence of the defining relations.
We will often work with a positive monoid M of the group F .It is defined by the monoid presentation that coincides with (1.1).The group F is the group of quotients of this monoid so that F = M M −1 .Elementary reasons show that any finite subset in F can be moved into M up to a right multiplication by an element g ∈ F (see Lemma 3.8 in Section 3).

Density of Cayley graphs
In [26] we constructed a family of finite subgraphs of the Cayley graph of F in the standard generating set {x 0 , x 1 }.The densities of these subgraphs approach 3.In the Addendum to the same paper we demonstrated a modification showing that the densities of finite subgraphs can strictly exceed 3.
An essential improvement was made by Belk and Brown [4,5].They gave a family of finite subgraphs depending on two integer parameters k ≥ 0 and n ≥ 1.Here the density of the corresponding subgraphs in the same Cayley graph approaches 3.5.
We need an explicit definition of these sets.Let BB(n, k) denote the set of all marked binary forests with n leaves, where all trees have height at most k.We regard it as a subset of the left Cayley graph of F in standard generators.Let us describe how the group generators act on the vertices (all actions are left partial ones).
The generator x 0 acts by shifting the marker one position left if this is possible.Action of x −1 0 means moving the marker one step to the right.The action of x 1 is as follows.If the marked tree is trivial, this is not applied.If the marked tree is T = (T 1 ˆT2 ), then we remove its caret and mark the tree T 1 .It is easy to see that applying x1 = x 1 x −1 0 means the same replacing T 1 by T 2 for the marked tree (notice that x 1 acts first).Now one can see that x 1 and x1 are totally symmetric.They generate F so one can regard them as the most natural generators besides the standard ones.
The action of x −1 1 and x−1 1 is defined analogously.Namely, if the marked tree of a forest is rightmost, then x −1 1 cannot be applied.Otherwise, if the marked tree T has a tree T ′′ to the right of it, then we add a caret to these trees and the tree (T ˆT ′′ ) will be marked in the result.Notice that if we are inside BB(n, k), then both trees T , T ′′ must have height < k: otherwise x −1 1 cannot be applied.As for the action of x−1 1 , it cannot be applied if T is leftmost.Otherwise the marked tree T has a tree T ′ to the left of it.Here we add a caret to these trees and the tree (T ′ ˆT ) will be marked in the result.As above, both trees T ′ , T must have height < k to be possible to stay inside BB(n, k).
Let us define a sequence of polynomials by induction: (2.2) Notice that Φ k (x) is the generating polynomial for the set of trees of height at most k.This means that the coefficient on x n in this polynomial shows the number of such trees with n leaves.This follows directly from (2.2).The summand x corresponds to the trivial tree (with one leaf); for the tree T = (T 1 ˆT2 ) of height ≤ k we have height ≤ k − 1 for each of the trees T 1 , T 2 .By induction, the pair of them has generating function Φ k−1 (x) 2 .This agrees with (2.2).
It is easy to see that the equation Φ k (x) = 1 has a unique positive root that we denote by ξ k .It is known that ξ k approaches 1  4 as k → ∞.It is not hard to see that for a random marked binary forest from BB(n, k), the probability to accept x 0 or x −1 0 approaches 1 as n → ∞.As for x 1 and x −1 1 , the probability for a random vertex in the automaton BB(n, k) to accept it approaches 1  4 for n ≫ k ≫ 1.The same holds for symmteric generators x1 and x−1 1 .
It follows from these remarks that the density of the Cayley graph of F in standard generators {x 0 , x 1 } is at least 3.5 since x ±1 0 are almost always accepted, and each of the x ±1 1 is accepted with probability close to 1 4 .In other terms, it follows that ι * (F, A) ≤ 1 2 for A = {x 0 , x 1 }.This was a remarkable result obtained by Belk and Brown in [4,5].
If we look from the same point of view to the symmetric generating set {x 1 , x1 }, then we see that the density of finite subsets BB(n, k) will approach 3 for n ≫ k ≫ 1.However, this is not the best estimate.
Let we have a marked forest of the form . . ., T −1 , T 0 , T 1 , . .., where T 0 is marked.Suppose that T 0 is a trivial tree and each of the neighbour trees T −1 , T 1 has height k.In this case no generator of the form x ±1 1 or x±1 1 can be accepted by such a vertex in the automaton.This means that such vertices are isolated in BB(n, k) as a subgraph of the left Cayley of F in the symmetric generating set.The event to get an isolated vertex holds with guaranteed probability p > 0, where p is a constant.Therefore, if we remove such vertices from the automaton, then the density will necessarily increase.Here is the main result from [27].Notice that this trick give nothing for increasing density in case of standard generating set.Indeed, almost all vertices in BB(n, k) accept both x 0 and x −1 0 .So if we remove a vertex isolated in the previous graph, we will need to remove 4 directed edges incident to it.This never has an effect of increasing density.
Let us mention one more result of the same paper.It concerns another symmetric set with the generator x 0 added.Notice that this fact was improved in our latest papers.Equivalently, the generating set S does not have doubling property, that is, there are finite subsets Y in F such that the 1-neighbourhood N 1 (Y ) = (S ±1 ∪ {1})Y has cardinality strictly less than 2|Y |.
In [32] we introduced the concept of an evacuation scheme on a Cayley graph.Let we have an infinite group G generated by a finite set A. Let Γ = C(G; A) be its Cayley graph (right or left).To each vertex v we assign an infinite simple path p v starting at v in the Cayley graph.Suppose that there exists a constant C such that each directed egde e can participate in these paths at most C times.In this case we say that the family (p v ) v∈G is an evacuation scheme on the Cayley graph Γ.
To be more precise: we claim that the total number of occurrences of each edge e in paths of the form p v (v ∈ G) does not exceed C.
Roughly speaking, the paths p v bring all verties to infinity.Without any restrictions, such an object always exists.However, if we claim that each edge participates in the scheme a uniformly bounded number of times, then we get the property equivalent to nonamenabilty.The following statement easily follows from known criteria.

Proposition 2.3 ([32]
).A group G with finite generating set A is non-amenable if and only if there exists an evacuation scheme on its Cayley graph.
Suppose that we have an evacuation scheme with constant K on the Cayley graph Γ = C(G; A).Let Y be a finite nonempty subset of G.We know that each path p v (v ∈ Y ) must leave Y at some step.Hence there exists an initial segment pv of p v that has its terminal point on ∂Y .This leads to the concept of an evacuation scheme with constant K on a finite subgraph.This is a collection of paths in Y of the form pv .This finite path starts at v and ends on the inner boundary ∂Y .For each edge e we claim that the total number of its occurrences in the paths pv (v ∈ Y ) does not exceed K.
Having an evacuation scheme on Y , we can assume that paths pv are simple (otherwise we can remove some loops).Also we can claim that if e occurs in the evacuation paths, then e −1 does not occur.Indeed, if pv = p 1 ep 2 , pu = p 3 e −1 p 4 , then one can replace these evacuation paths by p 1 p 4 , p 3 p 2 , respectively.

Proposition 2.4 ([32]
).Let G be an infinite group generated by a finite set A. An evacuation scheme on the Cayley graph Γ = C(G; A) exists if and only if there exist evacuation schemes on all its finite nonempty subgraphs.The constants for both cases are the same.
An important case of an evacuation scheme happens if C = 1 in the definition.In this case we say that we have a pure evacuation scheme on the Cayley graph.This means that each directed edge can participate in evacuation paths of the form p v at most once.We obtained a criterion for existing of such a scheme.

Theorem 2.5 ([32]
).Let G be an infinite group generated by a finite set A. A pure evacuation scheme on the Cayley graph Γ = C(G; A) exists if and only if the group G is strongly non-amenable with respect to A, that is, its Cheeger isoperimetric constant is at least one: The following statement improves one of our previous results.
According to Theorem 2.5, this means that there is no pure evacuation scheme on the Cayley graph of F in these generators.
The construction used in the proof of Theorem 2.5 was almost the same as the one from [27].It looked like the estimate 3.5 for the density in standard generators could not be exceeded.Many attepts to do that since 2004 led to an opinion that the construction by Belk and Brown was optimal.This was mentioned by Burillo in [9]; the author discussed this with Jim Belk on some conferences.In [32] we even formulated a conjecture that i * (F ; A) = 1 2 for the standard generating set A = {x 0 , x 1 }.However, it turned out that the conjecture was false.
In [33] we got an improvement of the above estimate.The main result sounds as follows.
Theorem 2.7 ( [33]).The density of the Cayley graph of Thompson's group F in the standard set of generators {x 0 , x 1 } is strictly greater than 3.5.Equivalenly, the Cheeger isoperimetric constant of F in the same set of generators is stricltly less than 1 2 .The basic idea of the proof is as follows.The subset BB(n, k) of the left Cayley graph of F has some fragments with small density.The probability to meet such a fragment has a positive uniform lower bound.If we remove such fragments from the subgraph, we increase its density exceeding the value 3.5.
Let us describe the fragments we are interested in.Let we have a rooted binary forest . . ., T 0 , T 1 , T 2 , T 3 , T 4 , . . ., where T 0 is marked and all trees have height at most k.We claim that trees T i (0 ≤ i ≤ 4) exist in this forest, and the following conditions hold: • T 0 and T 2 are trivial trees, • T 1 and T 3 have height k, • T 4 is a nontrivial tree.
The latter condition is added for simplicity.A forest satisfying the listed conditions is called special.To each of these forests we assign vertices a, b, c of the left Cayley graph of F in the standard generating set.Here a corresponds to the forest with T 0 as a marked tree; b and c denote the forests where T 1 and T 2 are marked trees, respectively.
In the left Cayley graph of F in standard generators these vertices look as in Figure 1.Since the tree T 0 is empty, we cannot remove a caret from it.This means that a does not accept x 1 in the automaton BB(n, k).The tree T 1 has height k so a caret cannot be added to T 0 and T 1 to stay within BB(n, k).So a can accept only letters x 0 and x −1 0 .Exactly the same situation holds for the vertex c.Notice that the leftmost edge labelled by x 0 may not belong to the subgraph if T 0 is the leftmost tree in the special forest.
As for the vertex b, we can remove the caret from T 1 so b accepts x 1 .However, it does not accept x −1 1 since the tree T 1 has height k and no caret can be added to T 1 and T 2 .Thus a, c have degree 2 in BB(n, k) and b has degree 3 in the same subgraph.
If we have another special forest with the corresponding vertices a ′ , b ′ , c ′ , then no coincidences of vertices can occur.The only case could be a ′ = c (or a = c ′ , which is totally symmetric).However, this is impossible by the choice of the tree T 4 in the special forest.If we go from a ′ by a path labelled by x −2 0 , then we meet vertex c ′ that corresponds to the trivial tree.Going along the path with the same label from c, we get the forest with marked tree T 4 , which is nontrivial by definition.So this condition allows us to avoid repetitions.3:11 Using properties of generating functions, we estimate the probability to meet a special forest.It turns out that the probability has a lower bound p = 1 1200 .This leads to finite subgraphs in the Cayley graph of F with density exceeding 3.5004.
The size of the set we deal with is very huge: we estimate is as 2 2 7200 .Recall that, according to [38], the size of Folner sets in F (provided it is amenable) has a very fast growth: as a tower of exponents.We think that Theorem 2.7 increases the chances for the group F to be amenable.
Our construction disproves one more conjecture from [32].We thought that the Cayley graph of F for the set {x 0 , x 1 , x 2 } of generators has density 5.In this case the Cheeger isoperimteric constant is 1 so there exists a pure evacuation scheme on the graph according to Theorem 2.5.However, the following fact is true.
Theorem 2.8 ( [33]).The Cheeger isoperimetric constant of the Cayley graph of Thompson's group F in the generating set {x 0 , x 1 , x 2 } is strictly less than 1.Equivalently, the density of the corresponding Cayley graph strictly exceeds 5.This means that there are no pure evacuation schemes on the Cayley graph of F in these generators.
The fragment of the left Cayley graph of F in generators {x 0 , x 1 , x 2 } will look as in Figure 2 for any special forest (we keep previous notation).Notice that trees T 1 , T 3 of the special forest have height k.So no carets can be placed over any pair of trees of the form T i , T i+1 , where 0 ≤ i ≤ 3.This explains why no edges labelled by x −1 1 , x −1 2 can be accepted by vertices in the picture.The vertex a corresponds to a trivial marked tree so it does not accept x 1 .However, it accepts x 2 since the tree to the right of T 0 has a caret.A similar situation holds for the vertex c.As for b, it accepts x 1 but it does not accept x 2 since the tree T 2 to the right of the marked tree T 1 is empty.
The condition on the tree T 4 allows us to avoid repetitions of vertices a, b, c for different special forests.Now we are subject to remove vertices of the form a, b, c for all special forests.We also remove geometric edges incident to these vertices.For 3 vertices we remove no more than 14 directed edges from the graph.The key point here is inequality 14  3 < 5.Here 5 is the limit of densities for BB(n, k) considered as subgraphs in the left Cayley graph of F in generators {x 0 , x 1 , x 2 }.In the previous Section the same rôle was played by inequality

Equations in group rings and their systems
Tamari [44] shows that if a group G is amenable, then the group ring R = K[G] satsfies the Ore condition for any field K.This means that for any a, b ∈ R there exist u, v ∈ R such that au = bv, where u ̸ = 0 or v ̸ = 0.This stament can be easily generalized.Suppose that instead of one linear equation au = bv with coefficients in R we have a system of them, where the number of variables exceeds the number of equations: Cardinality arguments based on the Følner criterion show that for amenable group G, this system always has a nonzero solution.
In [3] Bartholdi shows that the converse to the above statement is true.This gives a new criterion for amenabilty of groups.Although Theorem 1.1 in [3] concerns the so-called GOE and MEP properties of automata (Gardens of Eden and Mutually Erasable Patterns), the proof of it allows one to extract the following statement.
(i) G is amenable (ii) For any field K and for any system of m linear equations over R = K[G] in n > m variables, there exists a nonzero solution.
In the Appendix to the same paper, Kielak shows that if the group ring K[G] has no zero divisors, both properties are equivalent to the Ore condition.In particular, this holds for R. Thompson's group F .It is orderable, so there are no zero divisors in a group ring over a field.So we quote the following Theorem 3.2 ([3]).(Kielak) The group F is amenable if and only if the group ring K[F ] over any field satisfies the Ore condition.
Let M be a cancellative monoid.It is known from [44] that if M is left amenable then the monoid ring K[M ] satisfies Ore condition, that is, there exist nontrivial common right multiples for the elements of this ring.In [16] Donnelly shows that a partial converse to this statement is true.Namely, if the monoid Z + [M ] of all elements of Z[M ] with positive coefficients has nonzero common right multiples, then M is left amenable.He asks whether the converse is true for this particular statement.
In [28] we show that the converse is false even for the case of groups.Say, if M is a free metabelian group, then M is amenable but the Ore condition fails for Z + [M ].Besides, we study the case of the monoid M of positive elements of Thompson's group F .Notice that the amenability problem for the group F is equivalent to left amenability of the monoid M ; on the other hand, it is known that M is not right amenable [23].We show that for this case the monoid Z + [M ] does not satisfy Ore condition.That is, even if F is amenable, this cannot be shown using the above sufficient condition.
The proof was based on the following Lemma 3.3 ([28]).Let M be a monoid embeddable into a group G. Suppose that the monoid Z + [M ] has nonzero common right multiples.Then for any a, b ∈ M there exists a relation of the form a ±1 b ±1 . . .a ±1 b ±1 = 1 that holds in G.
On the other hand, we prove that this property does not hold if M is the free metabelian group with basis {a, b}.This implies Theorem 3.4 ( [28]).There exists a left amenable cancellative monoid M (actually, an amenable group) such that the monoid Z + [M ] does not satisfy the Ore condition.
The following property of the group F clarifies the situation for it with respect to Donnelly's condition.Lemma 3.5 ([28]).Let x 0 , x 1 , ... , x m , ... be the standard generating set for R. Thompson's group F .Then any word of the form w does not represent the identity element of F provided all i 1 , ... , i k are even and all j 1 , ... , j k are odd.
Here is the consequence of the above statements: Theorem 3.6 ( [28]).Let M be positive monoid of R. Thomspon's group F .Then Z + [M ] does not satisfy Ore condition.Now let us review the results from [29].First of all, the group F can be replaced by the monoid M in the statement of the Ore condition: This means the we can just forget about negative powers of variables working in the ring of skew polynomials satisfting x j x i = x i x j+1 for 0 ≤ i < j.This follows from an elementary fact that any finite subset in F can be sent into M under right multiplication.More precisely: Lemma 3.8.For any g 1 , ..., g n ∈ F there exists g ∈ F such that g 1 g, . . ., g n g ∈ M .
Another step is to reduce the Ore equation au = bv in K[M ] to the case when a, b are homogeneous polynomials of the same degree.Now we have a bunch of equations in K[M ] indexed by two parameters.The first one is d, the degree of homogeneous polynomials a and b.The second one is m, where m is the highest subscript in variables we involve.The general strategy can be the following: we try to solve as much equations in K[M ] as we can, using this classification.For a pair of numbers d ≥ 1, m ≥ 1, we can take a, b as linear combinations of monomials of degree d in variables x 0 , x 1 , ... , x m with indefinite coefficients.We can think about these coefficients as elements of the field of rational functions over K with a number of variables.
More precisely, we can state the general problem as follows.Any finite system of monomials of degree d ≥ 1 is contained in a set of the form S m+1,m+d+1 for some m ≥ 1.This set consists of all elements in the monoid M with normal forms x i 1 . . .x i d , where denote the set of all linear combinations of elements of S ⊂ M with coefficients in K. (Strange parameters in the notation we use is explained by the following reason: if we represent elements of M by positive semigroup diagrams, then the set of them will consists of (x m+1 , x m+d+1 )-diagrams.) ], find a nonzero solution of the equation au = bv, where u, v ∈ K[M ], or prove that it does not exist.
According to Theorem 3.2 by Kielak, and Lemma 3.9 on homogeneous equations, we have the following alternative.If the Problem P d,m has positive solution for any d, m ≥ 1 (that is, we can find nonzero solutions), then the group F is amenable.If this Problem has negative solution for at least one case, then F is not amenable.
We start with the case of polynomials of degree d = 1 for arbitrary m.Let us consider equations of the form We show that all of them have nonzero solutions in K[M ] for arbitrary coefficients α i , β i ∈ K (0 ≤ i ≤ m).This follows from cardinality reasons, see [29,Section 2].However, this approach gives us soultions of very high degree.More precisely, we prove the following This means that Problem P 1,m has positive solution for any m ≥ 1.However, the degree of u, v have quadratic growth with respect to m.This estimate can be essentially improved that will be shown later.Before doing that, we mention the following fact that follows from cardinality reasons.This means that Problem P 2,1 has positive solution.The cases that come after that are already unknown.This is d = 2, m = 2, where a, b are linear combinations of 9 monomials of degree 2: x 2 0 , x 0 x 1 , x 0 x 2 , x 0 x 3 , x 2 1 , x 1 x 2 , x 1 x 3 , x 2 2 , x 2 x 3 .This is one possible candidate to obtain the negative answer.If, nevertheless, this Problem P 2,2 has positive answer (that is, there exist nonzero solutions), then one can try Problem P 3,1 , where a, b are linear combinations of 14 monomials of degree 3: The further strategy can be as follows: starting from equations of a simple form, we try not only to prove they have nonzero solutions (which is not so hard), but also try to describe somehow the set of all their solutions.
Say, if we have equation of the form au = bv, where a = α 0 x 0 + α 1 x 1 , b = β 0 x 0 + β 1 x 1 , then the description of all its solutions is easy.Namely, u = (β 0 x 0 + β 1 x 2 )w, v = (α 0 x 0 + α 1 x 2 )w for any w ∈ R = K[M ].This means that the intersection of two principal right ideals aR ∩ bR is a principal right ideal.
We also know how to describe all solutions of the equation (α 0 x 0 + α 1 x 1 + α 2 x 2 )u = (β 0 x 0 + β 1 x 1 + β 2 x 2 )v.This will be done later.For this case, the description will be more complicated.In particular, the intersection aR ∩ bR is no longer a principal right ideal.
If instead of one equation we have a system of equations of the form for any k ≥ 2, then it also has a nonzero solution.Indeed, the product is left divisible by α i x 0 + β i x 1 for any 1 ≤ i ≤ k, which can be checked directly.A much more interesting example of a system of equations looks as follows.Let us state it as a separate problem.
Problem Q k : Given k +1 linear combinations of elements x 0 , x 1 , x 2 , consider a system of k equations with k + 1 unknowns: Find a nonzero solution of this system, where u 0 , u 1 , . . ., u k ∈ K[M ], or prove that it does not exist.
Notice that Q 1 has been already considered.To solve Q 2 in positive, it suffices to find a finite set Y with the property |AY | < 3  2 |Y |, where A = {x 0 , x 1 , x 2 }.This can be done by cardinality reasons similar to the ones used in the proof of Theorem 3.11.The estimate there will be also n ≥ 45.In fact, we are able to prove a much stronger fact.Namely, using the result of [27], we can construct a finite set Y with the property |AY | < 4  3 |Y |.The size of Y is really huge, it does not have transparent description.This immediately implies that Problem Q 3 has a positive solution.
Theorem 3.12 ( [31]).Let A = {x 0 , x 1 , x 2 }.Then there exists a finite subset S ⊂ F satisfying |AS| < 4  3 |S|.As an immediate corollary, we have Corollary 3.13 ([31]).Let R = K[F ] be a group ring of F over a field K.For any 4 linear combinations of elements x 0 , x 1 , x 2 with coefficients in K, the system of 3 equations with 4 unknowns [17] that F is non-amenable if and only if there exists ε > 0 such that for any finite set Y ⊂ F , one has |AY | ≥ (1 + ε)|Y |, where A = {x 0 , x 1 , x 2 } (see also [15]).For the set Y here, one can assume without loss of generality that Y is contained in S 4,n for some n.This gives some evidence that the amenability problem for F has very close relationship with the family of Problems Q k .The case k = 4 looks as a possible candidiate to a negative solution (that is, all solutions are zero).If true, this will imply that the constant ε = 1 4 fits into the above condition.Now we give an improved version of Theorem 3.10.
According to our strategy, we are interested in describing the set of all solutions to the above equation.This is easy to do for m = 1.Let m = 2.The equation from Theorem 3.14 can be rewritten as 2) up to a linear transformation, where α, β are some coefficients.We will assume both of them are nonzero: otherwise the description becomes trivial.
First of all, we take a solution of this equation extracted from the proof of Theorem 3.14.One can check directly that the following polynomials satisfy (3.2): We say that (u 0 , v 0 ) is a basic solution.Now we are going to show how to extract all solutions from it.
By M 1 we denote the submonoid of M generated by x 1 , x 2 , ... .We prove the following lemma using the same ideas as in the number-theoretical Remainder Theorem.

Lemma 3.15 ([29]). For any
Let us denote by ϕ an endomorphism of F that takes each x i to x i+1 (i ≥ 0).The description of the set of solutions for (3.2) can be reduced to the case when α = β.Namely, if (u, v) is a solution for (3.2) with d = deg u = deg v > 1, then the following equalities hold: where (u ′ , v ′ ) is a solution of the equation (3.2) with α = β, and deg u So the problem can be reduced to the case when α = β in the equation.For this case we have the following inductive decription.

Theorem 3.16 ([29]
).Let β ̸ = 0 be an element of a field K. Let us consider the equation Then for any its solution, one has the following presentation for its first unknown: where k ≥ 0, and w i belongs to K[M i ], where M i is the submonoid of M generated by x i , x i+1 , . . .(0 ≤ i ≤ k).3:17 Notice that if we know u, then the second unknown v is determined uniquely since the group ring K[F ] over a field has no zero divisors.Now let us review the results of [31] that continue the above research.We are going to consider equations and their systems in the group ring K[F ] of the group F over a field K. Ring coefficients of the equations will not be assumed to be homogeneous polynomials in Let we have a group word w = ξ 1 ξ 2 . . .ξ n where ξ Therefore, the above equation can be rewritten in the form (1 − x 0 )u = (1 − x 1 )v, where u, v ∈ K[F ].Elements u, v are defined uniquely by the relation w = 1 in F .
Recall that amenability of an m-generated group G can be characterized in terms of its cogrowth rate.Let A be a generating set for G, where |A| = m.Denote by P n the number of group words over A of length n representing the trivial element of G. Then G is amenable if and only if lim sup n→∞ n √ P n = 2m.This is a famous Kesten criterion [35,36].If Pn the number of reduced group words over A of length n trivial in G, then G is amenable if and only if lim sup n→∞ n Pn = 2m − 1.This is Grigorchuk criterion [22].
According to these remarks, we see that it is useful to know the description of all solutions to the equation (1 − x 0 )u = (1 − x 1 )v in the group ring of F .If we take one of the defining relations of F , namely, x  , and apply to it the above procedure, then we get to the following equation in the group ring of F : (1 − x 0 ) • (1 − x 1 )(1 + x 1 − x 2 ) = (1 − x 1 ) • (1 − x 3 − x 2 0 + x 0 x 1 ).We call the pair (u, v) a basic solution of the equation (1 − x 0 )u = (1 − x 1 )v, where u = (1 − x 1 )(1 + x 1 − x 2 ), v = 1 − x 3 − x 2 0 + x 0 x 1 .Now we are going to get all solutions in terms of the basic one.It suffices to present u in its general form since v is uniquely determined by u.Theorem 3.17 ([31]).Let R = K[F ] be a group ring of F over a field K. Let  The proof is based on the following fact that looks interesting in itself.The union of supports of the elements u 0 , u 1 , . . ., u m ∈ K[F ] is a multi-dimensional analog of a relation in F .An explicit form of this finite subset or even its size are unknown already for the case m = 2.
One more general criterion for non-amenability of a group can be presented in terms of group series.Let G be a group generated by a finite set A = {a 1 , . . ., a m }.For any field K we denote by K[[G]] the space of infinite sums of the form g∈G α(g) • g, where α(g) ∈ K are coefficients (g ∈ G).Obviously, G acts on the left and on the right on this space.These actions can be naturally extended to the group ring.This gives K[[G]] the structure of a K[G]-bimodule.For the application we need, we assume that K = R will be the field of reals (or rationals, if necessary).In partucular, this is applied to F , where we need an equality of the form

1. 2 .
Automata, density, and isoperimetric constants.The cardinality of a finite set Y will be denoted by |Y |.
For any set Y of vertices, by B r (Y ) we denote the r-neighbourhood of Y , that is, the union of all balls B r (v), where v runs over Y .By ∂ o Y we denote the outer boundary of Y , that is, the set B 1 (Y ) \ Y .An edge e is called internal whenever it connects two vertices of Y .If an edge e connects a vertex of Y with a vertex outside Y , then we call it external.That is, e connects a vertex in ∂Y with a vertex in ∂ o Y .The set of external edges form the Cheeger boundary of Y denoted by ∂ * Y .
. It is denoted by δ(Y ).A Cheeger isoperimetric constant of the subgraph Y is the quotient ι * (Y ) = |∂ * Y |/|Y |.It follows directly from the definitions that δ(Y ) + ι * (Y ) = 2m.Indeed, each vertex v has degree 2m in the Cayley graph Γ.This is the sum of the number of internal edges starting at v, which is deg Y (v), and the number of external edges starting at v. Taking the sum over all v ∈ Y , we have 2m|Y |, which is equal to v deg Y (v) + |∂ * Y |.Dividing by |Y |, we get the above equality.
) where a b = b −1 ab by definition.Also we define a commutator [a, b] = a −1 a b = a −1 b −1 ab and notation a ↔ b whenever a commutes with b, that is, ab = ba.

Theorem 2 . 2 .
For the symmetric generating set S = {x 0 , x 1 , x1 }, there exists finite subsets Y ⊂ F in the Cayley graph of Thompson's group F such that |∂Y | < |Y |.

1 Figure 1 :
Figure 1: The vertices a, b, c in the left Cayley graph of F in standard generators.

2 Figure 2 :
Figure 2: Fragment of the left Cayley graph of F in generators {x 0 , x 1 , x 2 } for any special forest.

Lemma 3 . 7 (
[29]).For any field K, the group ring K[F ] satisfies the Ore condition if and only if the monoid ring K[M ] satisfies the Ore condition.

Lemma 3 . 9 (
[29]).Suppose that any equation of the form au = bv has a nonzero solution in K[M ] provided a, b are homogeneous polynomials of the same degree.Then K[M ] satisfies the Ore condition.

Theorem 3 .
18 ([31]).Let R = K[F ] be a group ring of F over a field K. Then for any element b ∈ R, the equation (1 − x 0 )u = bv has a non-zero solution in R.

Theorem 3 .
19 ([31]).Let b ∈ K[M ] be any element in the monoid ring R = K[M ].Then the set B = {b, ϕ(b), ϕ 2 (b), • • • } is not a free basis of the right R-module it generates.Notice that we do not know whether any equation of the form (1 − x 1 )u = bv has a nontrivial solution.So we asked the following Question 3.20.Let R = K[F ] be a group ring of F over a field K.Is it true that for any element b ∈ R, the equation (1 − x 1 )u = bv has a non-zero solution in R? Using the automorphism x 0 → x −1 0 , x 1 → x1 = x 1 x −1 0 , one can reduce the above problem to the case of equation (x 0 − x 1 )u = bv, where b ∈ K[M ].We can extract from Theorem 3.18 the following Corollary 3.21 ([31]).For any m ≥ 1 there exists a non-zero solution to the system of equations (1 − x 0 )u 0 = (1 − x 1 )u 1 = • • • = (1 − x m )u m (3.4) in the group ring of F .
Proposition 1.1.A group G with finite set of generators A is amenable if and only if its Cheeger isoperimetric constant iz zero: ι * (G; A) = 0. Y |/|Y | = 0.Such subsets of vertices are called Folner sets.Informally, this means that almost all vertices of these sets are internal.