Group theory·OpenAI Math·Result 252·Lean formalisation listed
A torsion-free hyperbolic group that is neither residually finite nor linear over any field
One of the 372 results in OpenAI's public mathematics catalogue, filed under group theory: 1 preprint, the latest dated 23 September 2026. This page quotes OpenAI's own summary of the claim, links the paper and abstract, and embeds the animation.
In one line: OpenAI reports a result titled ‘A torsion-free hyperbolic group that is neither residually finite nor linear over any field’ (group theory) in a preprint dated 23 September 2026. It has not been peer reviewed. The channel's animation, Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More, is on YouTube.
Chapter 68 of 84 in Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · starts at 23:13 · watch from there on YouTube →
What OpenAI claims
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)
Paper
A torsion-free hyperbolic group that is not residually finite
OpenAI's abstract We construct a torsion-free word-hyperbolic group that is not residually finite, answering the residual-finiteness question for hyperbolic groups negatively.
Status
Preprint. The paper has not been independently reviewed. Treat the result as OpenAI's claim until other mathematicians have checked it. If a video exists, it illustrates the statement; it is not the proof.
Quick answers
- Is it proven?
- Not independently yet. OpenAI posted a preprint on 23 September 2026; it has not been peer reviewed, so treat the result as OpenAI's claim until other mathematicians have checked it.
- What exactly does OpenAI claim?
- 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)
- Where is the proof?
- In the paper, A torsion-free hyperbolic group that is not residually finite (PDF).
- Is there a video?
- Yes, on YouTube (29:10, chapter 68).
Also in
- Chapter 7 of 9 in Cannon's Conjecture, Thompson's Group F and 7 More in Group Theory · starts at 2:03 · watch from there on YouTube →
The video
Every OpenAI Math Result, Part 2 of 4: PCP for PPAD, Birkhoff Conjecture and 82 More · Compilation · 29:10 · chapter 68 from 23:13 · description, sources and chapters →
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