Why Thompson’s group F cannot have almost invariant finite sets

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Family 248 · Group theory
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-22

1. The question

The integers can be approximated by long finite intervals that barely change when shifted one step. Only the two endpoints are lost or gained, while the interval’s total size grows. Amenability asks for an analogous way to average over a group without favoring one location over another.

Thompson’s group $F$ is a famous test case. Its elements are simple-looking piecewise linear transformations of an interval, but the group resists several familiar classifications. It contains no nonabelian free subgroup, so one standard source of nonamenability is absent. Yet that absence does not guarantee amenability.

The manuscript in family 248 claims that $F$ is nonamenable. It constructs a fixed finite collection of translations for which every nonempty finite subset of $F$ has a definite relative boundary. No sequence of larger and larger sets can make all those boundary proportions vanish.

Its proposed argument connects three subjects: interval subdivisions, averaging in groups, and the geometry of infinite-dimensional Hilbert spaces.

2. Background and essential definitions

What is Thompson’s group F?

An element of $F$ is an increasing piecewise linear homeomorphism from $[0,1]$ onto itself. It has finitely many linear pieces, its breakpoints are dyadic rationals, and each slope is an integer power of two. A dyadic rational is a number of the form $k/2^r$, with integers $k$ and $r\geq0$.

For example, the following map belongs to $F$:

$$g(x)=\begin{cases} x/2,&0\leq x\leq1/2,\\ x-1/4,&1/2\leq x\leq3/4,\\ 2x-1,&3/4\leq x\leq1. \end{cases}$$

Its pieces fit continuously, it is increasing, and its slopes are $2^{-1},2^0,2^1$. Composition and inversion preserve the defining conditions. The manuscript uses $hg=h\circ g$ and treats $F$ as a discrete group.

A basic dyadic interval is one cell of a uniform dyadic grid, such as $[3/8,4/8]$. Finite partitions into such intervals give a convenient combinatorial model. Restricting a partition to one of its dyadic subintervals and stretching that interval back to $[0,1]$ produces a smaller normalized partition.

Almost invariant finite sets

For a finite subset $A$ of a group and an element $h$, write $hA=\{ha:a\in A\}$. The symmetric difference $hA\triangle A$ consists of points in exactly one of the two sets.

The Følner criterion says that a discrete group is amenable precisely when, for every finite set $S$ of translations and every $\varepsilon>0$, there is a nonempty finite $A$ with

$$\frac{|hA\triangle A|}{|A|}<\varepsilon \qquad\text{for every }h\in S.$$

For the additive group of integers, long intervals have this property for any fixed finite collection of shifts. Their boundaries grow much more slowly than their interiors.

An equivalent formulation uses an invariant mean: a positive normalized linear rule for averaging all bounded real functions on the group, unchanged by translation. It is not an ordinary countably additive uniform probability distribution on an infinite group.

Nonamenability means that some fixed finite set $S$ and some fixed positive tolerance defeat every finite candidate $A$. The order of these choices matters: the translations must not be chosen anew after seeing each candidate.

A Hilbert-space obstruction

A real Hilbert space has an inner product and the associated norm. Its closed unit ball is convex, so averages of its points remain inside it.

A map $f$ is Lipschitz if a constant $L$ satisfies $\|f(x)-f(y)\|\leq L\|x-y\|$. An approximate fixed point would be a point with small $\|f(x)-x\|$.

Benyamini and Sternfeld proved that an infinite-dimensional Hilbert ball admits a Lipschitz self-map whose displacement is uniformly positive:

$$\|f(x)-x\|\geq\delta>0 \quad\text{for every point }x\text{ in the ball}.$$

This does not contradict Brouwer’s fixed-point theorem, which concerns finite-dimensional compact balls. The infinite-dimensional ball is not compact in its norm topology.

3. Development history and previous partial results

Thompson introduced $F$ in 1965. Cannon, Floyd and Parry’s introductory account records Geoghegan’s 1979 conjecture that it is nonamenable. The question became a central example in the study of amenability because standard structural tests did not settle it.

Brin and Squier proved that $F$ contains no nonabelian free subgroup. Meanwhile, $F$ is not elementary amenable. The elementary amenable class is generated from finite and abelian groups by several natural closure operations, including extensions and directed unions. It is only a subclass of amenable groups, so being outside it is insufficient to establish nonamenability.

Brown and Geoghegan showed that $F$ has strong topological finiteness properties: it has a classifying space with finitely many cells in every dimension. Thus the difficulty is not simply that the group lacks a manageable finite description.

Moore studied possible Følner sets quantitatively and proved tower-type lower bounds on their sizes. Such bounds say that almost invariant sets, if they exist, must be extraordinarily large. They do not alone rule out their existence.

Moore also characterized amenability of $F$ through a scalar convex Ramsey property for finite ordered binary trees. This connected averaging over the group to finite combinatorial structures. Guba developed estimates for densities of finite Cayley subgraphs and restrictions on invariant means using diagram-defined classes.

Haagerup and Olesen found an operator-algebraic implication: simplicity of the reduced group operator algebra of Thompson’s group $T$ would imply nonamenability of $F$. This is a conditional route, not a deduction from the bare definition of $F$.

The manuscript also discusses earlier complete-resolution claims. It notes Akhmedov’s 2021 nonamenability claim, Shavgulidze’s amenability claim and Moore’s criticism of that approach, and a separately withdrawn amenability manuscript of Moore. The withdrawn work is distinct from Moore’s published Ramsey characterization. These are reported as part of the source’s historical context, without independently adjudicating their arguments.

The new proposed route uses finite averages of scalar correlations. It aims to turn approximate translation invariance into an approximate fixed point of a map specifically chosen to have none.

4. What the new manuscripts claim

The main theorem of the manuscript dated 23 September 2026 is simply that Thompson’s group $F$ is not amenable.

More concretely, the paper constructs a finite set $S\subset F$ and a positive constant $c$ such that every nonempty finite $A\subset F$ satisfies

$$\max_{h\in S}\frac{|hA\triangle A|}{|A|}\geq c.$$

The translations and the constant are fixed before $A$ is chosen. This is exactly the kind of uniform obstruction that contradicts the Følner criterion.

The source also records two consequences that use separate companion theorems. First, for every $\varepsilon>0$, it obtains a representation of $F$ by bounded invertible operators on a separable complex Hilbert space with operator norms at most $1+\varepsilon$, but with no bounded change of Hilbert coordinates making all operators unitary. This uses the unitarizability theorem discussed in family 251.

Second, it applies the percolation theorem of family 214 to every simple Cayley graph from a finite symmetric generating set of $F$. In Bernoulli bond percolation, each edge is independently retained with probability $p$. The source obtains a nonempty interval between the threshold for an infinite cluster and the threshold for a unique infinite cluster.

It also states the stronger operator bound from that companion and an interval on which infinitely many infinite clusters occur simultaneously in the usual coupled percolation process. These consequences are not inputs to the proof of nonamenability.

5. Why the advance matters

Importance: 5/5, provisional. The amenability problem for $F$ is a longstanding central question about one of the most intensively studied finitely presented groups. Its claimed resolution would settle a test case that evades the usual free-subgroup argument.

The result would give a specific, highly structured example where finite subsets cannot approximate translation-invariant averaging, even though the group contains no nonabelian free subgroup. Nonamenable groups without such subgroups were already known; the significance lies in resolving the question for this particular group and its rich interval and tree models.

The argument’s proposed mechanism also links an elementary-looking partition structure to an infinite-dimensional geometric obstruction. Its companion-dependent consequences connect the conclusion to representation theory and random connectivity on Cayley graphs.

6. The main idea

Recursively color partitions

Choose a Lipschitz map $f$ on a Hilbert unit ball with displacement at least $\delta$. Then choose $D$ separated internal dyadic intervals, called parents, and put an affine copy of this family inside every parent.

The manuscript assigns a Hilbert-ball color to each finite dyadic partition. If the partition does not respect every parent interval, its color is zero. Otherwise, restrict to the parents, normalize those restrictions, average their colors, and apply $f$.

This recursion terminates because each restriction has fewer cells than the original partition. It does not require solving a fixed-point equation in order to define the coloring.

For a sufficiently fine partition transformed by $g\in F$, let $X_i(g)$ be its parent colors and $Y_{ij}(g)$ its descendant colors. Set

$$m(g)=\frac1D\sum_iX_i(g),\qquad z_i(g)=\frac1D\sum_jY_{ij}(g).$$

The recursive rule gives the exact relation $X_i(g)=f(z_i(g))$. Thus $m(g)$ is the average of the $f(z_i(g))$.

Almost invariance compares scalar correlations

Elements of $F$ can carry any ordered pair of separated internal basic intervals affinely onto another such pair. The source chooses finitely many of these transports once and for all, forming the set $S$.

Given a candidate finite set $A$, it chooses one partition level fine enough for every element of $A$ and every translated set $hA$, with $h\in S$. This level may depend on $A$; the transport set does not.

If the relative boundaries are small, averages over $A$ barely change under these translations. Applying this elementary cancellation observation to scalar inner products makes the averaged correlations of separated interval colors nearly equal.

Not every pair is separated. A descendant is nested inside its own parent, and diagonal pairs repeat the same interval. Those exceptions occupy only a proportion of order $1/D$ among the relevant pairs.

An approximate fixed point becomes unavoidable

Expand the squared distance $\|z_i(g)-m(g)\|^2$ using inner products. The common correlation of separated pairs cancels. Unit-ball bounds control the exceptional pairs, while the translation error contributes the relative-boundary quantity

$$\eta=\max_{h\in S}\frac{|hA\triangle A|}{|A|}.$$

The manuscript derives a bound of size $D^{-1}+\eta$ for the averaged squared distance. Convexity and the Lipschitz property then transfer this to the displacement of $m(g)$ under $f$. Its stated estimate is

$$\delta^2\leq4L^2\bigl(D^{-1}+(1-D^{-1})\eta\bigr).$$

Choose $D$ large enough that the first term is too small to account for $\delta^2$. The inequality forces $\eta$ to stay positively bounded away from zero, independently of $A$.

The nonlinear map is handled through its Lipschitz estimate; the argument does not simply commute it with an average. The contradiction is that sufficiently good Følner sets would force displacement below the map’s known positive lower bound.

7. Scope, limitations, and open questions

The result concerns the discrete group $F$. It is not a general assertion that all groups of interval transformations are nonamenable, nor does it concern every possible topology on a homeomorphism group.

The boundary obstruction is given for a particular finite collection of translations. The paper does not claim an optimal expansion constant for the usual two-generator Cayley graph, or an exact random-walk spectral radius.

Nonamenability also does not produce a nonabelian free subgroup. The classical absence of such a subgroup in $F$ remains part of the problem’s distinctive structure.

The representation and percolation conclusions require their cited companion claims. They should not be read as independent consequences of elementary definitions alone, and they are not used to establish the main theorem.

Quantitative optimization of the transport set and boundary constant, and related amenability questions for other transformation groups, are separate from the stated qualitative resolution for $F$.

8. Sources and further reading

Cannon, Floyd and Parry, Introductory notes on Richard Thompson’s groups, L’Enseignement Mathématique 42 (1996), 215–256, gives the group models and historical conjecture. Brin and Squier, Groups of piecewise linear homeomorphisms of the real line, proves the relevant free-subgroup restriction.

Følner’s original criterion connects amenability to finite boundaries. Moore’s Følner-function lower bounds and Ramsey characterization give major earlier approaches.

Benyamini and Sternfeld, Spheres in infinite-dimensional normed spaces are Lipschitz contractible, provides the fixed-point obstruction. The source’s historical discussion also cites Akhmedov’s 2021 version and Moore’s note on Shavgulidze’s argument.

Source manuscripts

• M1. OpenAI, Thompson’s group F is nonamenable (September 23, 2026).

Sources consulted on 22 October 2026 at repository revision 1581d4c. Download the manuscript citations.