Result 252, Group theory

A torsion-free hyperbolic group that is neither residually finite nor linear over any field

Constructs a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question negatively. One fixed nonidentity element is killed by every finite-dimensional linear representation over every commutative field, so the group is not linear over any such field.

Lean formalization Disproof or counterexample

The bigger picture

Why it matters

Finite groups and matrices are standard ways to make abstract groups easier to study. This claimed construction shows that both can miss an element of a group even when its geometry is tightly controlled.

What changes?

The unreviewed manuscript reports a torsion-free word-hyperbolic group: every nonidentity element has infinite order, and its geometry has uniformly thin triangles. It is not residually finite, meaning finite group images cannot distinguish every nonidentity element from the identity. More strongly, the summary states that one fixed nonidentity element maps to the identity matrix under every finite-dimensional linear representation over every commutative field. Thus no faithful matrix representation exists, in any finite dimension over any such field.

What does that help mathematicians do?

The claimed example rules out the expectation that hyperbolic geometry alone guarantees detection of group elements through finite groups, even when elements of finite order are excluded. Its matrix obstruction is especially strong: increasing the dimension or changing the field never reveals the specified element. Researchers therefore cannot assume that finite images or finite-dimensional matrix representations retain all the information in a torsion-free hyperbolic group.

Are there practical applications?

The immediate value is foundational: the result would mark a boundary for studying hyperbolic groups through finite quotients and matrix representations. Arguments that require these representations to distinguish every element would need additional assumptions beyond torsion-freeness and hyperbolicity. The supplied material describes no practical algorithm or deployment.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

A torsion-free hyperbolic group that is not residually finite

September 23, 2026 21 pages Main result formalized in Lean

We construct a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question for hyperbolic groups negatively.

PDF LaTeX source
Cite (BibTeX) 
@misc{OAI:a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026,
  author = {{OpenAI}},
  title = {{A torsion-free hyperbolic group that is not residually finite}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026/paper.pdf}{OAI:a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026}},
  year = {2026}
}

Lean formalization

OpenAI's note on what the formalization covers, from lean/docs/252.md.

A torsion-free hyperbolic group that is neither residually finite nor linear over any field

The following describes the scope of the Lean formalization related to the following accompanying paper(s):

Scope

The residual-finiteness question for hyperbolic groups asks whether every nonidentity element survives in some finite quotient. The formalized result gives a negative answer by constructing a torsion-free word-hyperbolic group that is not residually finite.

Nonlinearity over every field is not a separate conclusion of the statement linked below.

Comparator links

Result Comparator statement
Torsion-free hyperbolic group that is not residually finite TorsionFreeHyperbolic.lean

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.