OpenAI research collection · 4 manuscripts · Source checked 2026-10-19
In familiar finite square matrices, if AB equals the identity, BA does too. This family constructs more exotic group-algebra settings where that expectation fails and develops related counterexamples.
The claims concern specialised infinite algebraic structures. They do not contradict ordinary finite-dimensional matrix arithmetic.
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.
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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.
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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∈ 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…
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∈ 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.
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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.
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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∈ K[G] with ab=1 but ba\ne1. The same elements define a cellular automaton on K^(G) that is…
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∈ K[G] with ab=1 but ba\ne1. The same elements define a cellular automaton on K^(G) that is injective but not surjective.
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