∑math20

Group theory · Result 252

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

1 paperLean

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.

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1

The problem

What is being asked, where it comes from, and why it is hard. No proof needed.

▶Introduction to the problemin preparation
2

The solution proposed by OpenAI

How the paper solves it, step by step.

Full explanation
▶The proposed solutionin preparation

Papers 1

First page of A torsion-free hyperbolic group that is not residually finite

A torsion-free hyperbolic group that is not residually finite

OpenAI · September 23, 2026

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

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Cite
@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}
}

What was formalized in Lean

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

ResultComparator statement
Torsion-free hyperbolic group that is not residually finiteTorsionFreeHyperbolic.lean

Scope note on GitHub →