Kaplansky’s conjectures fail for torsion-free hyperbolic groups in positive characteristic

Mghirbi, Nidhal; Douglas, Seth

doi:10.5281/zenodo.23873781

Results · Groups and their finite shadows

Kaplansky’s conjectures fail for torsion-free hyperbolic groups in positive characteristic

Nidhal Mghirbi and Seth Douglas · version 1.0.3 · Zenodo, 1 November 2026 · 52 pages · doi:10.5281/zenodo.23873781 · PDF

In plain wordsFour algebra conjectures, the oldest from 1940, fail in every prime characteristic, even for the well-behaved hyperbolic groups where they looked safest.

The resultat three depths

In one sentenceFor every prime $p$ there is a single torsion-free word-hyperbolic group whose group algebra, over any field of characteristic $p$ , has an element with a one-sided inverse that is not two-sided; that one failure breaks all four of Kaplansky’s conjectures at once.

As the paper states it, in briefFor every prime $p$ , one finitely presented, torsion-free, word-hyperbolic group whose group algebra over every field of characteristic $p$ is not directly finite and has a non-trivial idempotent, a zero divisor and a non-trivial unit.

For a scientist
One paragraph

Higman and Kaplansky asked whether the group algebra $K [G]$ of a torsion-free group has only trivial units, no zero divisors and no idempotents other than 0 and 1, and whether every group algebra is directly finite: Higman’s 1940 thesis contains the first two, and Kaplansky posed the problems in 1957, 1970 and 1972. Gardam disproved the unit conjecture over $𝔽_{2}$ in 2021, Murray extended this to every positive characteristic and Gardam to the complex numbers; in 2026 OpenAI disproved the zero-divisor and direct-finiteness conjectures, and S. V. Ivanov extended zero divisors to every odd characteristic. For torsion-free hyperbolic groups the conjectures looked robust, and no such group was known to violate any of the four. The paper shows that in positive characteristic all four fail: for $p = 2$ , OpenAI’s group is hyperbolic with probability $1 - o (1)$ ; for odd $p$ the groups are new, built from OpenAI’s direct-finiteness construction with a signed cancellation, found first by Ivanov and independently by the authors, an exact solution of a colour-balance problem and a rational Perron certificate. Hyperbolicity comes from one deterministic criterion: a pattern-free labelled graph, coned off, gives an aspherical group with a linear isoperimetric inequality. The Main Theorem, for every prime, is formalized in Lean 4 by Nidhal Mghirbi. In characteristic zero the unit and zero-divisor conjectures remain open for these groups.

For an expert
The theorems, as stated in the paper

Main Theorem

For every prime $p$ there are a finitely presented, torsion-free, word-hyperbolic group $G$ with a finite two-dimensional classifying space and elements $a, b \in 𝔽_{p} [G]$ with $a b = 1 \neq b a$ . Consequently, for every field $K$ of characteristic $p$ , the group algebra $K [G]$ is not directly finite, and it contains an idempotent other than 0 and 1, a zero divisor, and a unit of order $p$ that is not of the form $λ g$ .

Theorem B

If $Γ$ is $(ε, d, λ)$ -pattern-free at some scale $L$ , then the complex $X$ is aspherical, no path in $Γ$ from a root to a different vertex has trivial label in $G = π_{1} (X)$ , and every word $w$ that is trivial in $G$ satisfies $Area (w) \leq \frac{16}{13 ε} | w |$ .

Theorem C

For every odd prime $p$ there are a finitely presented, torsion-free, word-hyperbolic group $H_{p}$ with a finite two-dimensional classifying space and elements $a, b, c \in 𝔽_{p} [H_{p}]$ with $a b = 1$ , $a c = 0$ and $c \neq 0$ . Hence $b a \neq 1$ , and for every field $K$ of characteristic $p$ the group algebra $K [H_{p}]$ is not directly finite and contains an idempotent other than 0 and 1, a zero divisor and a unit of order $p$ that is not of the form $λ g$ . Moreover, $H_{p}$ is neither sofic nor surjunctive.

The Main Theorem comes from Theorem A and Corollary 5.2 for $p = 2$ , where the group and the elements are OpenAI’s and what is new is hyperbolicity with probability $1 - o (1)$ , and from Theorem C for odd $p$ . By the paper’s Table 1, in characteristic 0 units and zero divisors are open for torsion-free hyperbolic groups, and idempotents and direct finiteness hold.

The question

“For a field K and a group G, Higman and Kaplansky asked whether, when G is torsion-free, the group algebra K[G] has only the trivial units […], no zero divisors […], and no idempotents other than 0 and 1 […]; and whether, for every group G, K[G] is directly finite […]. Higman’s 1940 thesis contains (U) and (Z); Fox asked (Z) for integral group rings; and Kaplansky posed the problems.”

The paper, Section 1, citing Higman (1940), Fox (1953) and Kaplansky (1957, Problem 6; 1970; 1972)

“Let F be a field, G a group. The group algebra A(G) of G is the set of all finite linear combinations of group elements with coefficients in F, with the natural operations of addition and multiplication. Problem 6. If G has no elements of finite order, does A(G) have no divisors of 0?”

I. Kaplansky, Problems in the theory of rings, Report of a Conference on Linear Algebras, NAS–NRC Publication 502 (1957), Section 4, Problem 6, p. 2: the zero-divisor problem as printed. The restriction to hyperbolic groups is the paper’s own question.

Questionson the index

Extended Do Kaplansky’s group-ring conjectures hold for every torsion-free group? posed by others
Partly answered Do Kaplansky’s group-ring conjectures hold for torsion-free hyperbolic groups?

How this was checked

Blind audits: Two blind audits before version 1.0.2.
Independent computation: Exact certificates on more than one engine
Lean: The main theorem and its consequences, for every prime, formalized in Lean 4 by Nidhal Mghirbi; source and recorded axioms in the repository. Lean source
Peer review: Not yet.

Each check, what it covers, and what it leaves to the written proof.

Each check, what it covers, and what it leaves to the written proof.
Check	Covers	Does not cover
Lean 4	The Main Theorem and its consequences over every field of characteristic p, for every prime, by Nidhal Mghirbi, including existence forms of the random-graph estimates it uses; standard axioms only, confirmed on 1 November 2026 by an independent fresh build of the released files and two independent kernel replays (Lean’s leanchecker and the nanoda checker). The statement file copies its definitions from OpenAI’s public formal challenges.	The probability statement of Theorem A, non-soficity and non-surjunctivity, Connell’s theorem (used for the units of Corollaries 5.3 and 5.4 and Proposition 6.9), the explicit area constants, Section 6 apart from Lemma 6.1, Corollary 7.13, Section 8 and Theorem D. That the formal definitions mean what the paper means is a point of trust.
Verification scripts	Finite arithmetic and sample instances: the bicyclic identities, the constants, the projective-plane designs for p = 3 on two further independently written constructions, exact Perron row bounds for odd p up to 199; recorded outputs in `verify/`.	The proofs. The scripts check instances; they do not replace the arguments.
Blind audits	Two blind audits before version 1.0.2, refutation-first, of the paper, the release files and the Lean source, by readers that had seen none of the earlier reviews.	Errors shared by every reader. An audit is evidence, not a guarantee.
Peer review	Not yet.	A preprint on Zenodo.

Every audit here is internal. The repository’s DEPENDENCIES.md maps every statement to its inputs, and Appendix C of the paper maps each result to its Lean declaration. How we check results.

Label checked 1 November 2026.

Novelty and credit

Novelty: Extends OpenAI’s 2026 constructions, credited; their hyperbolicity is new. S. V. Ivanov (arXiv:2610.17662) has precedence for torsion-free zero divisors in odd characteristic, found independently by us. Bunina, Lee, Miasnikov and Muliarchyk (arXiv:2610.22536) independently gave finitely presented groups that are not directly finite in odd characteristic, without torsion-freeness or hyperbolicity.

Higman (1940 thesis) and Kaplansky (1957, 1970, 1972) posed the problems; Fox asked the zero-divisor question for integral group rings.
The cone-on-graphs construction, the random models and their estimates, the parity certificates, and the groups of Corollaries 5.2 to 5.4 and Theorems A and D are OpenAI’s; so is the accounting that turns a bounded counting estimate into a linear isoperimetric inequality, used by OpenAI for a different random complex.
The signed cancellation, the two-colour model and the transfer of OpenAI’s estimates to it were found first by S. V. Ivanov and independently by the authors; Ivanov has precedence. The sign rule appears earlier as Mineyev’s signature condition.
Bunina, Lee, Miasnikov and Muliarchyk independently gave finitely presented groups that are not directly finite in every odd characteristic; torsion-freeness and hyperbolicity are not claimed there. Building on the first version of the paper, Miah gives explicit finite incidence data for one-sided inverses in odd characteristic.
The Lean formalization is Nidhal Mghirbi’s; it builds on OpenAI’s Lean development.

Recordreleases, corrections and independent work

14 Oct 2026Version 1.0.0.doi:10.5281/zenodo.23374264
14 Oct 2026Correction. Version 1.0.1 gives the author order correctly: Nidhal Mghirbi, Seth Douglas.doi:10.5281/zenodo.23374592
15 Oct 2026Bunina, Lee, Miasnikov and Muliarchyk, arXiv:2610.22536: groups that are not directly finite in every odd characteristic, independently, without torsion-freeness or hyperbolicity.
26 Oct 2026Correction. Version 1.0.2: Nidhal Mghirbi’s Lean 4 formalization of the Main Theorem; a sentence of the introduction corrected (Agol’s theorem gives the zero-divisor conjecture for cubulated hyperbolic groups, not for all cubulated groups); references updated. No mathematical statement changes.doi:10.5281/zenodo.23673698
1 Nov 2026Version 1.0.3: an independent rebuild and kernel replay of the Lean formalization recorded; the PDF attached to the Zenodo record; two wording changes (the scope of Kaplansky’s Problem 6; the universe of the fields in the Lean statement). No mathematical statement changes.doi:10.5281/zenodo.23873781

Release dates are Zenodo publication dates. S. V. Ivanov’s arXiv:2610.17662, which has precedence for torsion-free zero divisors in odd characteristic, is credited in every version.

Versionsfour, each with its own DOI

Every archived version, newest first. Dates are Zenodo publication dates. All versions: 10.5281/zenodo.23374263, which always resolves to the newest.
Version	Date	DOI	Notes
1.0.3	1 Nov 2026	10.5281/zenodo.23873781	Records an independent rebuild and kernel replay of the Lean formalization, attaches the PDF to the Zenodo record, and makes two wording changes (the scope of Kaplansky's Problem 6; the universe of the fields in the Lean statement). No mathematical statement changes. PDF hosted here.
1.0.2	26 Oct 2026	10.5281/zenodo.23673698	Correction: A sentence of the introduction attributed the zero-divisor conjecture to all cubulated groups; Agol’s theorem gives it for cubulated hyperbolic groups. No mathematical statement changes.
1.0.1	14 Oct 2026	10.5281/zenodo.23374592	Correction: Version 1.0.0 gave the author order wrongly. Version 1.0.1 lists Nidhal Mghirbi, Seth Douglas.
1.0.0	14 Oct 2026	10.5281/zenodo.23374264

Paper, code and links

PDF

papers/kaplansky-v1.0.3.pdf, version 1.0.3, 773,542 bytes, 52 pages.

SHA-256 9554a9f1b9eaae1086cd6ca75f8dabcee675759c0c6c796770b743ffbe22b547

Of paper/main.pdf, 773,542 bytes, attached to the Zenodo record and in its archive; identical in the repository at that commit. The same file is hosted here: papers/kaplansky-v1.0.3.pdf.

DOI

10.5281/zenodo.23873781; all versions 10.5281/zenodo.23374263

Code

github.com/Apsiape/kaplansky-conjectures-hyperbolic-groups

Lean

Lean source

Commit

70af652adcd6037445851b916f806058074bba62

The commit archived on Zenodo (Apsiape/kaplansky-conjectures-hyperbolic-groups, tag v1.0.3). The main branch has since recorded the DOI.

Verify

From a clone of the repository, with Python 3 and numpy, scipy, sympy, mpmath and networkx, as the README gives them:

sh verify/run_all.sh
python tools/check_abstract.py

The first reruns every script and overwrites its recorded output, so git diff verify/ shows any change. For the Lean development, with elan installed, in lean/KaplanskyLean/ (Lean v4.34.1, Mathlib d13f23b):

lake exe cache get
lake build
lake env lean Audit.lean

Lean, in detail

The Main Theorem and its consequences, for every prime: Kaplansky.Final.mainTheorem_proof and Kaplansky.Final.mainTheoremConsequences_proof; the build record of 25 October 2026 reports #print axioms = [propext, Classical.choice, Quot.sound] for both, and an independent rebuild of the published files on 1 November 2026 reproduced this, with kernel replays by leanchecker and nanoda.

Licence

Paper and documentation CC BY 4.0; scripts MIT or Apache 2.0, at your option; Lean files Apache 2.0, with OpenAI’s vendored modules under their Apache License 2.0.

Page checked against the paper, the release repository and the Zenodo record on 1 November 2026.

Corrections

26 Oct 2026Version 1.0.2. A sentence of the introduction attributed the zero-divisor conjecture to all cubulated groups; Agol’s theorem gives it for cubulated hyperbolic groups. No mathematical statement changes.doi:10.5281/zenodo.23673698
14 Oct 2026Version 1.0.1. Version 1.0.0 gave the author order wrongly. Version 1.0.1 lists Nidhal Mghirbi, Seth Douglas.doi:10.5281/zenodo.23374592

All corrections, and what counts as one.

Cite

BibTeX built from the repository’s CITATION.cff (preferred citation).

@misc{mghirbi2026kaplansky,
  author  = {Mghirbi, Nidhal and Douglas, Seth},
  title   = {Kaplansky's conjectures fail for torsion-free hyperbolic groups in positive characteristic},
  year    = {2026},
  version = {1.0.3},
  doi     = {10.5281/zenodo.23374263},
  url     = {https://doi.org/10.5281/zenodo.23374263}
}

kaplansky.bib All papers as BibTeX