An infinite finitely presented simple amenable group

One infinite group that is simple, amenable and described by finitely many rules at once.

1 paper September 23, 2026 Explained by AI

Three traits of infinite groups matter here. Finitely many rules describe them, they cannot be broken into smaller pieces, and they allow fair averages. Groups with the last two traits were known since 2013, but none with all three. This paper proves such a group exists, with a Lean formalization listed. The group moves pieces of polygons around inside a finite set of squares.

First step

2013

Kate Juschenko and Nicolas Monod, whose work gave infinite finitely generated simple amenable groups

Significance

Major

Whether a finitely presented infinite simple amenable group exists was a recorded open question in group theory.

Trust

Formal proof listed

Field

Group theory

1 paper

The explanation

Take the integers and the window D = {1, 2, …, 100}. Shift the window by 1. You get {2, …, 101}. The old and new windows differ in only two numbers: 1 and 101. So the change is 2 out of 100, or 2%. A longer window makes the share smaller. This is the Følner condition: for any finite set of moves and any ε > 0, some finite set D barely changes under each move. Groups with this property are called amenable. They allow fair averages.

A group is simple when its only normal subgroups are the trivial group and the whole groupSource 1. A normal subgroup is one that conjugation by any element leaves unchanged. It is finitely presented when finitely many generators and rules describe itSource 1. The question asked whether an infinite group can have all three properties togetherSource 2.

Partial answers were known. Juschenko and Monod proved in 2013 that full groups of minimal Cantor systems are amenableSource 3. With earlier work of Matui, this gave infinite finitely generated simple amenable groupsSource 4. Finite presentation was the property left outSource 5. Matui showed the groups from minimal subshifts are not finitely presentedSource 6. There is a deeper reason. Those groups embed locally into finite groups, so a finitely presented one would be residually finiteSource 7. Residually finite means every nonidentity element survives in some finite image. An infinite simple group cannot be residually finiteSource 7. Any nontrivial map to a finite group would have to be injective, which is impossible for an infinite group.

The paper proves there is an infinite finitely presented simple amenable groupSource 8. The group is an alternating subgroup of a polygon exchange group on finitely many squaresSource 9. A polygon exchange cuts squares into polygons and slides the pieces by translations. The polygon edges lie on lines in four directionsSource 10. Their levels use the golden ratio τ = (1 + √5)/2 ≈ 1.618. The obstacle was amenability in two dimensions. Elek and Monod built a two-dimensional Cantor system whose full group contains a free group, which is not amenableSource 11. Here, random polygon partitions that move almost unchanged under each generator give the averagesSource 12. A homology argument and a relation-propagation step give finite presentation.

How sure is this? A computer proof checker called Lean has a formal proof that there is one group that is infinite, finitely presented, simple and amenable. It does not include the paper's later general claims. No specialist review of the paper has been published.

What it doesn't show. This does not bound the number of generators or rules, and Lean does not cover the general claims about central kernels and enlargements.

Where it came from

This result comes from an unreleased OpenAI model, published in the openai/math release on 6 October 2026.

Hype check

Major if it holds, and the existence theorem has a Lean formalization. The general claims across families are not in the Lean note. No specialist review is published.

Words used

Amenable group: A group with finite sets that barely change under each move; equivalently, it allows fair averages.
Følner condition: For every finite set of moves and every ε, some finite set changes by less than an ε share under each move.
Simple group: A nontrivial group whose only normal subgroups are the trivial group and itself.
Finitely presented: Described by finitely many generators and finitely many defining rules.
Residually finite: Every nonidentity element stays nonidentity in some finite image of the group.
Cantor system: A continuous map acting on a Cantor set, a totally disconnected space like the infinite binary sequences.
Polygon exchange group: A group of maps that cut regions into polygons and move each piece by a translation.

The original paper

An infinite finitely presented simple amenable group (PDF, opens in a new tab)
September 23, 2026 Formal proof listed

Lean notes

All quoted lines (12)

It is simple if its only normal subgroups are the trivial group and $G$ , and finitely presented if $G\cong F(S)/\langle\!\langle R\rangle\!\rangle$ for finite sets of generators $S$ and relator words $R$ .
An infinite finitely presented simple amenable group Back to text
We address the existence problem asking whether these three properties can hold together in an infinite group.
An infinite finitely presented simple amenable group Back to text
Juschenko and Monod then proved amenability of the full group of every minimal homeomorphism of a Cantor space.
An infinite finitely presented simple amenable group Back to text
Combined with this earlier structure theory, their Corollary B gives infinite finitely generated simple amenable groups.
An infinite finitely presented simple amenable group Back to text
Finite presentation is the obstruction left by those examples.
An infinite finitely presented simple amenable group Back to text
Matui proved that the derived groups arising from minimal subshifts are not finitely presented.
An infinite finitely presented simple amenable group Back to text
An infinite simple group cannot be residually finite, since a nontrivial homomorphism from it to a finite group would be injective.
An infinite finitely presented simple amenable group Back to text
There exists an infinite finitely presented simple amenable group.
An infinite finitely presented simple amenable group Back to text
The group will be an alternating subgroup of a polygon exchange group on a sufficiently large finite number of squares.
An infinite finitely presented simple amenable group Back to text
The polygons have edges on level lines of $x,\quad y,\quad y-\lambda x,\quad x-\lambda y,$ at levels in $\OO=\ZZ[\tau]$ , where $\tau=(1+\sqrt5)/2$ and $\lambda$ is a sufficiently large power of $\tau$ .
An infinite finitely presented simple amenable group Back to text
Amenability does not follow from the rank of the acting group: Elek and Monod constructed a free minimal Cantor $\ZZ^2$ -action whose full group contains a nonabelian free group.
An infinite finitely presented simple amenable group Back to text
Matching cells of the same shape then yields almost-invariant measures on the exchange group.
An infinite finitely presented simple amenable group Back to text

Other sources

openai/math, Lean scope note for family 253 The formalized result answers the existence question affirmatively: there is one group that is infinite, finitely presented, simple, and amenable.
openai/math release notes (2026): model provenance and review status manuscripts and supporting proof artifacts produced by an internal OpenAI model

252A torsion-free hyperbolic group that is neither residually finite nor linear over any fieldGroup theory
254Classifying spaces and geometric obstructions for Artin groupsGroup theory
251Amenability, unitarizability, and strong Ulam stabilityGroup theory
255Quasi-isometric recognition of virtually polycyclic groupsGroup theory
250Boone–Higman embeddings with higher finitenessGroup theory

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