An infinite simple amenable group with a finite presentation

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Family 253 · Group theory
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-22

1. The question

Can an infinite group be simple, amenable, and finitely presented at the same time? Each adjective describes a different kind of structure. Simplicity forbids decomposition through normal subgroups. Amenability supplies finite sets that are almost unchanged by translation. Finite presentation says that finitely many generators and finitely many relations describe the entire group.

There are many infinite groups with two of these features. Making all three coexist has been substantially harder. The manuscript claims such a group, constructed from exchanges of polygonal pieces in finitely many squares. Its achievement is therefore about the compatibility of algebraic rigidity, averaging, and a finite description.

An especially striking consequence concerns finite approximations. An infinite simple group has no nontrivial homomorphism to a finite group. The claimed example nevertheless has the finite, almost invariant subsets required by amenability. These are different notions of approximation: finite subsets inside a group need not produce finite quotient groups.

2. Background and essential definitions

Finite rules and simplicity

A presentation

$$G=\langle s_1,\ldots,s_r\mid R_1,\ldots,R_t\rangle$$

constructs a group from words in generators and their inverses, with the displayed relations and their consequences imposed. It is a finite presentation when both lists are finite. A finitely generated group can require infinitely many relations; this is exactly the distinction that matters here.

A subgroup $N$ is normal if conjugation by any group element preserves it. A nontrivial group is simple if its only normal subgroups are the identity subgroup and the whole group. Kernels of homomorphisms are normal. Consequently, a homomorphism from an infinite simple group to a finite group must be trivial: otherwise it would be injective, which is impossible.

A group is perfect if it equals its commutator subgroup, generated by elements $aba^{-1}b^{-1}$. Every nonabelian simple group is perfect. Perfectness will also help turn a finite presentation of a related group into a finite presentation of the desired example.

Averaging without a finite quotient

For a discrete group, amenability can be expressed by the Følner condition. For every finite subset $K\subset G$ and every $\varepsilon>0$, there is a nonempty finite $D\subset G$ such that

$$\frac{|gD\mathbin{\triangle}D|}{|D|}<\varepsilon \qquad(g\in K).$$

Here $\triangle$ means symmetric difference. Thus multiplication by any of the finitely many prescribed elements changes only a small fraction of $D$. The sets $D$ need not be subgroups, and their size may grow rapidly as the requested error decreases.

For example, large boxes work for the integer lattice under addition. The construction in this manuscript is considerably less direct: its elements use different translations on different polygonal pieces. The fact that individual translations commute does not make a group of piecewise translations automatically amenable.

Exchanging polygonal pieces

An interval exchange cuts an interval into finitely many pieces and translates them into a new order. Polygon exchanges use two-dimensional pieces instead. One specifies the allowed polygons and translations so that compositions and inverses remain in the same class.

The paper uses finitely many copies of a square, called tracks. Its alternating subgroup is generated by local even permutations of disjoint translated copies of the same polygon. A three-cycle, which cyclically moves three such pieces, is the basic example. The number of tracks is fixed once and for all; the final group is not a union obtained by continually adding tracks.

3. Development history and previous partial results

The historical route runs through dynamics on a Cantor set, a compact space with many subsets that are simultaneously open and closed. Given a minimal homeomorphism, its topological full group consists of maps that agree locally with integer powers of that homeomorphism. Minimality means that every orbit is dense.

Giordano, Putnam and Skau developed the structural setting, and Matui identified simple commutator subgroups and strong finite-generation properties. In the minimal-subshift setting, these commutator groups are finitely generated, but Matui also proved that they are not finitely presented. Thus the missing finite presentation was already a concrete obstruction, not merely an omitted calculation.

Juschenko and Monod proved amenability of full groups of minimal Cantor systems. Combined with the simplicity results, this produced infinite finitely generated simple amenable groups. Their 2013 theorem settled the finite-generation problem, leaving the finite-presentation question addressed here.

Grigorchuk and Medynets established strong local approximation properties for these groups. In particular, being locally embeddable into finite groups means that every finite portion of the multiplication table can be reproduced in some finite group. A finitely presented group with this property is residually finite: its nonidentity elements can be separated in finite quotients. That is incompatible with infinite simplicity. The new construction must therefore escape an obstruction built into an important earlier source of examples.

Further constructions produced simple amenable groups with other remarkable features. Nekrashevych obtained finitely generated simple periodic groups of intermediate growth, and Kionke and Schesler obtained two-generated simple amenable groups. These developments do not supply the finite presentation claimed here, nor do their additional properties automatically transfer to the new example.

Moving from one-dimensional dynamics to polygon exchanges offers more geometric freedom, but creates an amenability problem. Elek and Monod showed that a topological full group associated with a free minimal action of $\mathbb Z^2$ can contain a nonabelian free group. An abelian acting group alone is therefore insufficient. Related work on higher-dimensional full groups, Penrose-type constructions, and rectangle exchanges supplies context without already giving the required combination of properties.

4. What the new manuscripts claim

The September 23, 2026 manuscript claims the existence of an infinite finitely presented simple amenable group.

More specifically, it constructs a polygon-exchange group $F_m$ and its alternating subgroup $A_m$, for one sufficiently large finite number $m$ of tracks. It claims that $F_m$ is amenable and that

$$A_m=[F_m,F_m]$$

is infinite, simple and finitely presented. Amenability passes to subgroups, giving the remaining property of $A_m$.

Two intermediate statements are important to interpreting finite presentation. The paper claims that the second integral homology group $H_2(A_m;\mathbb Z)$ is finitely generated and that there is a finitely presented central extension

$$1\longrightarrow K\longrightarrow \widehat G \longrightarrow A_m\longrightarrow 1.$$

A central extension means that every element of the kernel $K$ commutes with every element of $\widehat G$. The second homology of a perfect group is also called its Schur multiplier; roughly, it measures possible central extensions. Finite generation of that multiplier alone would not prove finite presentation. The manuscript combines it with the separately constructed finitely presented central extension.

All these properties concern a fixed group. Stabilization by adding tracks is used as a tool in the argument, but the theorem does not merely describe an infinite union of groups with progressively better properties.

5. Why the advance matters

Importance: 5/5, provisional. If correct, the claim resolves a central existence problem about simple amenable groups by adding the finite-presentation property that earlier constructions lacked.

Finite presentation changes what it means to have an example. Infinite families of local permutation relations can define a group precisely, yet still fail to reduce to finitely many rules. The manuscript claims that a finite collection of relations controls all the geometric configurations needed for simplicity and amenability.

The example would also sharpen the separation between averaging and finite quotient approximation. Its amenability supplies Følner sets, while infinite simplicity eliminates all nontrivial finite quotients. It therefore cannot be residually finite. This does not conflict with amenability: the two properties ask for different ways to approximate an infinite object.

Methodologically, the claim combines arithmetic polygon geometry, probability, group homology, and explicit relations. Its significance lies in making these ingredients work together for one group, rather than obtaining each desired property in a different construction.

6. The main idea

Two views of the same arithmetic

Let

$$\tau=\frac{1+\sqrt5}{2},\qquad \mathcal O=\mathbb Z[\tau],\qquad \lambda=\tau^a,$$

where $a$ is a sufficiently large positive integer. The ring $\mathcal O$ consists of numbers $u+v\tau$ with integers $u,v$. The allowed polygon edges lie on level lines of four functions:

$$x,\qquad y,\qquad y-\lambda x,\qquad x-\lambda y,$$

with levels in $\mathcal O$. Translations by vectors in $\mathcal O^2$ preserve this collection of lines. Boundaries are handled through a precise splitting convention, so pieces touching an edge do not create an ambiguity in composition.

The second real embedding sends $\tau$ to

$$\overline\tau=\frac{1-\sqrt5}{2}.$$

Since $|\overline\tau|<1$, making $\lambda$ large in ordinary geometry makes its conjugate $\overline\lambda$ small. This expanding-and-contracting pair of viewpoints is used both to organize polygon subdivisions and to control induction on their arithmetic data.

Local three-cycles can move pieces through spare regions. An area-comparison argument provides enough room to implement these moves. The manuscript uses them to establish perfectness, simplicity and infinitude of the alternating group.

Making random partitions almost invariant

To prove amenability, the paper starts with finitely many polygon exchanges and seeks finite probability distributions on group elements that are almost unchanged by those exchanges.

It constructs random polygon partitions adapted to the regions on which each prescribed exchange is a single translation. Two correlated random fields control subdivision points and selected boundary segments. Their multiscale correlations, organized through the conjugate embedding, make bounded translations change the distributions only slightly while keeping fluctuations under control.

On a high-probability event, cells respect the translation regions. Translating a cell therefore produces a cell of the same shape. Randomly matching cells of each shape converts the partitions into almost invariant probability distributions on the group. The Følner formulation then supplies the finite sets required for amenability.

The difficult issue is compatibility across boundaries. This is why a collection of abelian translations is only the starting material, not an amenability proof by itself.

From infinitely many configurations to finitely many relations

Finite presentation is handled through the central extension rather than by immediately listing every relation in $A_m$.

The paper chooses finitely many generators, including track permutations and balanced translations, whose translation vectors sum to zero. It imposes finitely many true relations to obtain $\widehat G$. Coordinate-cut relations are first controlled in a bounded arithmetic range. Inclined edges then let the argument separate small rectangles in ordinary geometry, while contraction in the conjugate coordinate keeps the induction within previously controlled arithmetic ranges.

Rectangle decompositions extend these laws to general allowed polygons. Exact conjugation rules eventually show that an element acting trivially on all tracks commutes with every generator of $\widehat G$. This identifies the kernel as central; it does not yet discard the kernel.

A separate homology argument proves finite generation of $H_2(A_m;\mathbb Z)$. It uses complexes of polygon embeddings, stabilization, and the fact that the allowed geometry has finitely many relevant corner types up to translation. A fixed reserve of tracks brings the needed low-degree information back to the chosen finite $m$.

Finally, the homology exact sequence for the central extension relates $H_2(A_m;\mathbb Z)$, $K$, and the abelianization of $\widehat G$. Both outer pieces needed to control $K$ are finitely generated, so $K$ is finitely generated as an abelian group. Adding finitely many kernel generators as relators to the finite presentation of $\widehat G$ gives the claimed finite presentation of $A_m$.

7. Scope, limitations, and open questions

The theorem is an existence statement with a specific arithmetic construction. It does not assert that arbitrary polygon-exchange groups are amenable or finitely presented. Restricting the allowed slopes, levels and translations is essential to the stated method.

The number of tracks is finite but chosen sufficiently large for several parts of the argument. The headline does not identify an optimal number of generators, an optimal presentation size, or a practical algorithm for finding efficient Følner sets.

The manuscript’s theorem should also be separated from properties of earlier examples. It does not by its headline claim make this group periodic, torsion-free, two-generated, or of intermediate growth. Those are distinct questions requiring additional information.

Finally, the local approximation obstruction remains instructive: this infinite simple finitely presented group cannot be locally embeddable into finite groups. Understanding its approximation properties therefore requires more than transferring the methods used for earlier Cantor full groups.

8. Sources and further reading

For the earlier simple-group constructions, see Matui’s study of Cantor minimal systems, Juschenko–Monod’s amenability theorem, and Grigorchuk–Medynets on local approximation. Elek–Monod’s two-dimensional example explains why amenability cannot be inferred from the translation group alone.

Nekrashevych’s intermediate-growth construction and Kionke–Schesler’s two-generator construction describe different refinements of the earlier existence results. The source manuscript supplies the specific arithmetic geometry and central-extension argument summarized here.

Source manuscripts

• M1. OpenAI, An infinite finitely presented simple amenable group (September 23, 2026).

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