Residual Finiteness of Hyperbolic Groups

Residual Finiteness of Hyperbolic Groups

OpenAI's claim

OpenAI presents a counterexample 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 are TorsionFreeHyperbolic.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.

Residual Finiteness of Hyperbolic Groups

Definitions and mathematical statement

A group $G$ is residually finite if its finite quotients separate its elements from the identity: \[ \forall g\in G\setminus\{1\}\ \exists F\text{ finite}\ \exists\varphi:G\to F: \varphi(g)\ne1. \] The conjecture asks for this property for every word-hyperbolic group. A different quotient may be used for each $g$, and no algorithm or quantitative bound on $|F|$ is required. Hyperbolicity is defined using a finite generating set and the thin-triangle condition on its Cayley graph.

Formulation and concept references: [S1].

Short English statement

Can every nonidentity element of every hyperbolic group be detected in some finite quotient?

Research attempt

Current frontier.

The supplied paper constructs residually finite hyperbolic groups that are not linear, so nonlinearity is not an obstruction to residual finiteness. [S1] Alekseev--Thom's 2026 manuscript obtains local embeddability in finite groups for Kazhdan groups with a specified sofic embedding and an ergodic centralizer action. It does not furnish those hypotheses for every hyperbolic group. [E2] Exact local embeddings must be distinguished from approximate permutation actions. For finitely presented groups the former condition is already equivalent to residual finiteness, as explained in Bradford, Section 2.3. [E1]

Attack route.

Complete partial generator actions on a large Cayley ball to permutations of a finite set. This works for free groups without an arithmetic representation. For a hyperbolic presentation, the missing step is exact simultaneous repair of relators without collapsing a chosen word. We prove the precise local criterion and test both completion and a counterexample strategy based on a concrete non-residually-finite one-relator group.

Attempt: a finite-presentation reduction.

Let $G=\langle S\mid\mathcal R\rangle$ with finite $\mathcal R$, relator lengths at most $L$, and symmetric word balls $B_R$. Suppose a finite group $F$ admits an injection $\iota:B_R\to F$ satisfying \[ \iota(xy)=\iota(x)\iota(y)\quad(x,y,xy\in B_R). \] For $R\ge\max\{L,|g|,1\}$ the generator assignment extends to a homomorphism $G\to F$ detecting $g\ne1$. Indeed $\iota(1)=1$ by idempotence, and inverse pairs are respected. All prefixes of every relator lie in $B_R$, so repeated local multiplication evaluates the relator to one. The presentation gives the homomorphism, and the same prefix calculation sends $g$ to $\iota(g)\ne1$.

Conversely, residual finiteness supplies a finite quotient detecting every nonidentity element of the finite set $B_R^{-1}B_R$: take a product of the individual finite quotients. This quotient is injective on $B_R$. Thus exact local embeddings for every $R$ are equivalent to the target for a fixed hyperbolic group, which is finitely presented. This is a self-contained verification of the standard reduction, not a weaker conjecture solved in place of residual finiteness. [E1]

Free completion and its relator defect.

For $F_d$, take the finite set of reduced words $B_R$. Right multiplication $v\mapsto vs$ is a partial bijection where both endpoints lie in $B_R$. Complete it to a permutation $P_s$ for each positive generator; unmatched domains and ranges have equal size. Set $P_{s^{-1}}=P_s^{-1}$. With the usual right-action convention these define an action of the free group, because there are no relations to check. Every reduced word $w$ of length at most $R$, read from the identity, follows its original path and ends at $w$. The resulting finite permutation quotient detects all such words and has order at most $|B_R|!$. The companion program constructs and checks these actions on every word through its tested radius.

Arbitrary completion fails in the presence of relations. Even for the radius-one ball $\{0,\pm x,\pm y\}$ of ${\mathbb{Z}}^2$, the completions \[ P_x=(-x\ 0\ x)(y\ {-y}),\qquad P_y=(-y\ 0\ y)(x\ {-x}) \] retain the prescribed partial translations but do not commute: $P_xP_y(0)=-y$ and $P_yP_x(0)=-x$, using rightmost-first composition in this displayed calculation. Enlarging the ball forces relators only on paths remaining in the interior, not on the newly completed boundary.

Nor is the boundary necessarily negligible. For $d\ge2$, \[ |B_R|=1+2d\frac{(2d-1)^R-1}{2d-2},\qquad \frac{|B_R\setminus B_{R-L}|}{|B_R|} \longrightarrow1-(2d-1)^{-L}. \] A fixed-thickness boundary can have positive limiting proportion. This invalidates a particular truncation estimate, not soficity or residual finiteness of these groups.

The exact repair statement needed is: for each hyperbolic presentation and nontrivial $g$, find finite permutations satisfying every defining relator identically while moving some basepoint along $g$. Hyperbolicity controls fillings of infinite Cayley-complex loops; no passage from that fact to the required exact finite permutation identities is proved here. Small Hamming error would not meet this requirement.

A failed counterexample transfer.

Consider \[ B=\langle a,b\mid ba^2b^{-1}=a^3\rangle, \qquad w=[bab^{-1},a]. \] Every map from $B$ to a finite group kills $w$. If the image of $a$ has order $m$, equality of orders of $a^2$ and $a^3$ forces $\gcd(m,2)=\gcd(m,3)=1$. Therefore both powers generate $\langle a\rangle$ and conjugation by $b$ normalizes this cyclic group. Thus $bab^{-1}$ commutes with $a$.

But $w=bab^{-1}ab a^{-1}b^{-1}a^{-1}$ is nontrivial by HNN normal form: none of its potential pinches has exponent divisible by the required $2$ or $3$. The base element $a$ has infinite order. This is the standard Baumslag--Solitar normal-form framework recalled in [E3]. It does not give a hyperbolic counterexample. In a word-hyperbolic group an infinite-order element has positive stable translation length, invariant under conjugacy and satisfying $\tau(a^k)=|k|\tau(a)$. The relation would force $2\tau(a)=3\tau(a)$, impossible. Thus the very relation producing this finite-quotient obstruction violates the loxodromic-element property of hyperbolic groups.

A hyperbolic group merely surjecting onto $B$ does not inherit this failure. The free group on $a,b$ already surjects onto $B$, while its residual finiteness was proved by the finite-ball construction. A quotient map or Rips-type surjection cannot by itself transfer the obstruction to its source.

Stress test.

Finite presentation enters at the relator-prefix step; no equivalence of LEF and residual finiteness is asserted for arbitrary infinitely presented groups. Exact identities are essential, not a vanishing fraction of violations. The example $B$ supplies an explicit nontrivial element killed by every finite quotient, but fails hyperbolicity. The 2026 centralizer criterion is useful only after its special sofic embedding and ergodicity are established; neither is assumed for the target groups. [E2]

Outcome.

Outcome: Exploratory approach with unresolved gap. A finite-ball separation construction for free groups and the exact obstruction to its naïve extension to presentations have been supplied. A concrete Baumslag--Solitar counterexample transfer has also been excluded. No new residual-finiteness theorem is claimed. The remaining step is relation-preserving finite completion, or construction of a genuinely hyperbolic group with an element invisible in all finite quotients.

Sources

• S1 Nicolas Tholozan and Konstantinos Tsouvalas, “Residually Finite Non-Linear Hyperbolic Groups,” Commentarii Mathematici Helvetici 100 $2025$, 1–9, DOI 10.4171/CMH/581. https://ems.press/content/serial-article-files/49988. Catalogue formulation source.
• E1 H. Bradford, Quantifying local embeddings into finite groups, Journal of Algebra 608 $2022$, 214--238; arXiv:2104.07111, Section 2.3. https://arxiv.org/html/2104.07111v2.
• E2 V. Alekseev and A. Thom, Centralizers of sofic approximations of Kazhdan groups, arXiv:2608.05362, 5 August 2026; ergodic-centralizer criterion. https://arxiv.org/abs/2608.05362.
• E3 M. Clay, M. Forester and J. Louwsma, Stable commutator length in Baumslag--Solitar groups and quasimorphisms for tree actions, arXiv:1310.3861, Section 2. https://arxiv.org/pdf/1310.3861.