Residual Finiteness and Minimal Presentations of Hyperbolic Groups
Status Open · unsolvedmath.comLLM status Open unverified
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Definitions and mathematical statement
Shared notation.
$\mathbb N$, $\mathbb Z$, $\mathbb Q$, $\mathbb R$, and $\mathbb C$ denote the natural numbers, integers, rationals, reals, and complex numbers, respectively. All additional parameters and hypotheses are specified in the question.Statement recorded in the supplied source.
Is every hyperbolic group residually finite? Separately, what is the smallest total relator length in a presentation of an infinite hyperbolic group with Kazhdan property (T)?Formulation references: [S1], [S2].
Short English statement
Is every hyperbolic group residually finite? Separately, what is the smallest total relator length in a presentation of an infinite hyperbolic group with Kazhdan property (T)?Sources
- S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30004458 (OWR-1703863-007). Catalogue attribution under CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
- S2 Original problem formulation and mathematical context. https://doi.org/10.4171/owr/2020/16.
Statement licence: CC-BY-SA-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2462
The main example refutes the residual-finiteness subquestion. It does not minimize the presentation length in the independent property-(T) problem. This entry records partial support; complete coverage of the question is not established by this catalogue.
Residual Finiteness and Minimal Presentations of Hyperbolic Groups
OpenAI's claim
OpenAI presents a mathematical result relevant to this question in the selected manuscript. A torsion-free hyperbolic group is claimed not residually finite and consequently not linear over any field.Mathematical statement of the claimed result
Result 1.
There is claimed to be a torsion-free word-hyperbolic group $G$ whose finite residual is nontrivial: \[ R_f(G)=\bigcap_{\substack{Q\text{ finite}\\\phi:G\to Q}}\ker\phi\ne\{1\}. \] For this finitely generated example, the linear-obstruction corollary states \[ \forall K\text{ commutative field},\ \forall n\ge1,\ \forall\rho:G\to\operatorname{GL}_n(K),\qquad R_f(G)\subseteq\ker\rho. \] Thus a single nonidentity element is annihilated by every such finite-dimensional representation.Outline of the argument
Punctured-line blocks with few short cycles give triangular relations meeting a geometric filling criterion. Real and prime-field rank comparisons produce a persistent finite-quotient obstruction, while local factorization certificates and planar metric bounds establish hyperbolicity and exclude torsion.Formal verification
For family 252, the selected example is torsion-free hyperbolic and fails residual finiteness. Nonlinearity is a further group-theoretic consequence, not a separate selected conclusion. The listed Comparator files areTorsionFreeHyperbolic.lean. Named declarations include OAI.Release075.main. 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
- OpenAI, A torsion-free hyperbolic group that is not residually finite, 2026-09-23; citation key
OAI:a-torsion-free-hyperbolic-group-that-is-not-residually-finite-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main; Statement source: cor:nonlinear. - Family 252: Lean scope.
- Comparator TorsionFreeHyperbolic.json.
- Comparator TorsionFreeHyperbolic.lean.
- OpenAI catalogue entry.