A torsion-free group algebra that is not directly finite

Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽₂ is not directly finite, disproving Kaplansky's conjecture even without torsion. Companion examples give injective nonsurjective cellular automata on all configurations, refuting Gottschalk's surjunctivity conjecture. Another counterexample is an integral group-ring matrix, invertible over the rational group ring, with Fuglede–Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture. (Lean)

Manuscripts (4)

A Torsion-Free Group Algebra That Is Not Directly Finite

October 4, 2026 · source · cite as \cite{OAI:A-Torsion-Free-Group-Algebra-That-Is-Not-Directly-Finite-October-4-2026}

We construct a finitely presented torsion-free counterexample to Kaplansky's direct finiteness conjecture over the field of two elements. The group admits a finite two-dimensional classifying complex.

A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Characteristic-Two-September-23-2026}

We disprove Kaplansky's direct-finiteness conjecture by constructing a finite field K of characteristic two, a finitely presented group G, and finite sums a,b\in K[G] with ab=1 but ba\ne1. The group G is nonsofic. The same elements define a cellular automaton on K^G that is injective but not surjective, disproving Gottschalk's surjunctivity conjecture.

A Counterexample to the Group-Ring Determinant Conjecture

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:A-Counterexample-to-the-Group-Ring-Determinant-Conjecture-September-23-2026}

We disprove the unrestricted group-ring Determinant Conjecture. We construct a finitely generated group G and a square matrix over \mathbb Z[G] that is invertible over \mathbb Q[G] and has Fuglede–Kadison determinant strictly between zero and one. The logarithmic integral defining the determinant is finite.

A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Odd Characteristic

September 26, 2026 · source · Lean formalization available · cite as \cite{OAI:A-Counterexample-to-Kaplanskys-Direct-Finiteness-Conjecture-in-Odd-Characteristic-September-26-2026}

We construct a counterexample to Kaplansky's direct-finiteness conjecture in odd characteristic. For one specified odd prime p, we obtain a field K of order p⁴, a finitely generated group G containing torsion, and finite sums a,b\in K[G] with ab=1 but ba\ne1. The same elements define a cellular automaton on K^G that is injective but not surjective.

Result 197 · Algebra. Imported from openai/math, a collection of manuscripts produced by an OpenAI model; results are at different stages of verification. Discussion below is by Lemma members.