Family 197AlgebraLeanNot yet digestedfeatured

A torsion-free group algebra that is not directly finite

Constructs a finitely presented torsion-free nonsofic group whose group algebra over 𝔽

_{2}

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.

Manuscripts in this family · 4

A Torsion-Free Group Algebra That Is Not Directly Finite

Oct 4, 2026·direct-finiteness.pdf

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.

Full abstract

PDF on GitHub LaTeX source

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

Sep 23, 2026·paper.pdfLean

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.

Full abstract

PDF on GitHub LaTeX source

A Counterexample to the Group-Ring Determinant Conjecture

Sep 23, 2026·paper.pdfLean

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.

Full abstract

PDF on GitHub LaTeX source

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

Sep 26, 2026·paper.pdfLean

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 p

^{4}

, 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.

Full abstract

PDF on GitHub LaTeX source

OpenAI also released the model's reasoning trace for this result: reasoning trace (PDF).