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Math-1287 · mathematics · Group theory

Kourovka Notebook Problem 21.146

Status Open · unsolvedmath.comLLM status Open unverified

Statement

Definitions and mathematical statement

A classifying space for a group $H$, denoted $K(H,1)$, is a connected CW-complex whose fundamental group is $H$ and whose higher homotopy groups vanish. Equivalently its universal cover is contractible. Its $n$-skeleton is the union of all cells of dimension at most $n$.

A group is of type $F_n$ if it has some classifying space with a finite $n$-skeleton. Thus $F_1$ is equivalent to finite generation and $F_2$ to finite presentability. Type $F_\infty$ means type $F_m$ for every positive integer $m$, equivalently the existence of a classifying space with finitely many cells in each dimension. This does not require the space to have finite dimension.

Fix any integer $n\ge3$. The first question asks whether every group $H$ of type $F_{n-1}$ embeds into a group $G$ of type $F_n$. The stronger question asks whether every such $H$ embeds into some group $G$ of type $F_\infty$.

In both formulations an embedding is an injective homomorphism $H\hookrightarrow G$, and the overgroup may depend on $H$ and on $n$. No finite-index or normality condition on the image is imposed. The request is for each $n\ge3$ and every group meeting the stated hypothesis, not just for groups already satisfying a higher finiteness property. The two targets distinguish a single improvement of skeleton finiteness from simultaneous finiteness in all dimensions.

Short English statement

Can every group of type $F_{n-1}$, for $n\ge3$, be embedded into one of type $F_n$, or even one of type $F_\infty$?

Sources

  • S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2655 (KOU-21.146), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
  • S2 The Kourovka Notebook, 21st edition, version 47 (30 September 2026), new problems, Problem 21.146. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.

Catalogue entry

Statement licence: CC-BY-4.0 · Source file

Contributions 1

Math-1287-A1 · Literature report · LLM-generated · unknown ·

OpenAI claim summary for Math-1287

For $n\ge3$, every group of type $F_{n-1}$ is finitely presented. The universal-group manuscript claims that one group of type $F_\infty$ contains all finitely presented groups. It therefore covers the requested embedding into $F_n$ and the stronger embedding into $F_\infty$, with a single overgroup independent of the input and of $n$. The simple-overgroup companion applies only to inputs with decidable word problem and is not needed for either unrestricted clause here.

Kourovka Notebook Problem 21.146

OpenAI's claim

The linked manuscripts present the following claims by OpenAI: One group of type $F_\infty$ is claimed to contain every finitely presented group, with finitely generated subgroups exactly the recursively presented ones; A group with finitely many generators and a decidable identity-word test is claimed to embed into a nontrivial simple group of type $F_\infty$. Together they give a claimed affirmative answer in the matching scope.

Mathematical statement of the claimed result

Result 1.
The claimed group $H$ has type $F_\infty$, meaning a classifying space with finite skeleton in every dimension, and satisfies \[ \forall G\text{ finitely presented},\qquad\exists\iota:G\hookrightarrow H. \] The corollary characterizes the finitely generated subgroups embeddable in this same $H$: \[ G\text{ finitely generated}:\qquad G\hookrightarrow H \quad\Longleftrightarrow\quad G\text{ recursively presented}. \] This does not assert embedding of every countable group.
Result 2.
Consider a group $G$ generated by finitely many elements, with an algorithm deciding its word problem. The assertion supplies a nontrivial simple group $H$ of type $F_\infty$ and an embedding \[ \iota:G\hookrightarrow H. \] Type $F_\infty$ requires finite skeleta of a classifying space in all dimensions. The decidability condition on $G$ is part of this theorem.

Outline of the argument

A universal finite presentation is inserted into faithful finite diagram envelopes using two systems of idempotents. An ascending-torus construction supplies uniform fillings with finitely many simplex types in each dimension, establishing the claimed higher finiteness of the universal envelope.

Encoded finite diagrams carry faithful actions with controlled stabilizers. A homological device for ascending tori provides fillings in every dimension, and stabilizers of finite type in a highly transitive action assemble the simple overgroup while retaining higher finiteness.

Formal verification

For family 250, the documented entries cover Boone--Higman equivalence, simple $F_\infty$ overgroups for decidable word problems, and one universal $F_\infty$ group. The converse characterization of all its recursively presented subgroups is extra manuscript scope. The listed Comparator files are BooneHigman.lean, SimpleOvergroups.lean, UniversalFInfinity.lean. Named declarations include OAI.UniversalFInfinity.universal_group_of_type_FInfinity. This site has not executed the Lean code. The scope descriptions do not certify the remaining manuscript arguments.

Status caveat

OpenAI attributes these manuscripts to work by an internal model. Its release cautions that arguments without formalization can contain mistakes. This catalogue has not independently checked the proofs; the labels report the mathematical claims attributed to OpenAI.

Sources

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