Ulam’s question answered
Every Lie group is a group of permutations of a countable set (Monod & Thom; key construction from GPT-6 Astra)
Event confirmedAwaiting review
Importance: major (4 of 5)The takeaway
Nicolas Monod and Andreas Thom prove that every Lie group embeds in the permutation group of a countable set, answering a classical question of Ulam. They say the key deformation was first built by GPT-6 Astra over three weeks of discussions; Thom earlier criticised how OpenAI credited human work.
Status
- Claim
Event confirmedAwaiting review
- Our reporting
- High confidence
- Verification
- Preprint only
- Importance
- Major (4 of 5)
- Last verified
- 14 October 2026
Your AI and this story
- GPT-6 Astra166 days after its cutoff
- Claude Opus 5.5105 days after its cutoff
- Gemini 4 Argon196 days after its cutoff
- Grok 4.7135 days after its cutoff
None of these four assistants can know about it. The closest, Claude Opus 5.5, stops 105 days before it.
Key facts
- Paper: arXiv 2610.18464 [math.GR], 13 Oct 2026, by Nicolas Monod and Andreas Thom
- Theorem A: every Lie group (second countable) is countably representable, i.e. embeds in Sym(N). Corollary B: every locally compact second countable group is countably representable
- History in the paper: Schreier and Ulam showed in 1935 that R is countably representable; Ulam asked whether all Lie groups are [Ula60, II.7; Ula64, V.2]. Linear Lie groups were known (Thomas 1999; Kallman 2000; Ershov–Churkin 2004), and Conversano–Monod (J. Algebra 2024) reduced the problem to connected simple Lie groups. The remaining obstacle was non-linear central extensions
- Key new ingredient (Theorem C): a countable-index subalgebra A of R with a continuous one-parameter family of ring homomorphisms from the inclusion back to a countable real-closed field, which makes the cohomological obstruction from central extensions vanish
- AI disclosure: ‘The key countable-index deformation was first constructed by OpenAI’s GPT 6 Astra during extended mathematical discussions. The argument was subsequently developed, checked and substantially rewritten by the authors’
- Acknowledgments: ‘The proof emerged from three weeks of discussions with OpenAI’s GPT, initially aimed at exploring SL3(R) with ring deformations and K-theoretic arguments. GPT Astra eventually produced the countable-index deformation underlying Theorem C. We then suggested extending the argument to SLn(R) and other simple Lie groups’
- Prior art the authors found while checking: ‘substantial parts of the argument already appear, notably the algebraic-analytic construction of Parusiński–Rond [PR20]’; the simultaneous Puiseux embedding with prescribed residues (Lemma 2.4) and its combination with countable-index gluing ‘appear to be new and creative’
- Context: in September Thom argued that OpenAI’s announced non-sofic group result relied on his 2019 work with Gábor Kun and that OpenAI’s attribution was misleading; this paper credits Astra openly
What happened
On 13 Oct 2026 Nicolas Monod and Andreas Thom posted arXiv 2610.184641, proving that every Lie group can be realised as a group of permutations of a countable set. Schreier and Ulam had shown this for R in 1935, which led Ulam to ask whether it holds for all Lie groups. Linear Lie groups were known, and Conversano and Monod reduced the question to connected simple Lie groups in 2024. The remaining case was simple groups that are only central extensions of linear groups, where the cohomological obstruction lives on “wild” subgroups of countable index.
The new tool is Theorem C: a subalgebra of R of countable index together with a continuous path of ring homomorphisms from a countable real-closed field to the inclusion, built from Puiseux series over fields of analytic functions and glued along countably many pieces. The continuity lets the authors trivialise the central extension on the corresponding subgroups.
The authors give a detailed account of the AI’s role. The proof “emerged from three weeks of discussions with OpenAI’s GPT”, which started on SL3(R). “GPT Astra eventually produced the countable-index deformation underlying Theorem C.” The humans proposed the extension to SLn(R) and other simple groups, checked and rewrote everything, and found that a large part of the construction already appears in Parusiński–Rond (2020). They still call the Puiseux embedding with prescribed residues and its gluing “new and creative”.
Andreas Thom is the Dresden group theorist who, in September, criticised OpenAI’s framing of Astra’s non-sofic group result as ignoring his work with Gábor Kun (see Astra’s ten advances). His recent Mathstodon posts did not mention this paper (checked 14 Oct). No Hacker News thread was found (Algolia, 14 Oct).
Why it matters
It is a classical, well-known question answered with a clearly credited AI contribution to the decisive step, and with an unusually frank account of what was new and what turned out to be in the literature. It comes from a mathematician who had publicly challenged AI-maths attribution, which makes the credit given here more significant.
What is disputed or not yet verified
| Verification | Unrefereed preprint by two established group theorists; no formal proof |
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Sources
2 sources from 2 sites. Numbers match the chips in the text.
2 sources: 1 primary, 1 reaction
Primary
Reactions
- Andreas Thom (guest post on Tao’s blog, Sept 11): On the existence of non-sofic groupsterrytao.wordpress.com, discussion
Changes
- Filed from the arXiv paper (arXiv AI-disclosure list)