A torsion-free group algebra that is not directly finite. Constructs a finitely presented torsion-free nonsofic group whose group algebra over š¯”½2 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 Fugledeā€“Kadison determinant strictly between zero and one, disproving the unrestricted Determinant Conjecture.
released 2026-10-04 | 3 theorems · 9 lemmas · 15 proofs · 15,047 words |
PLAY LEVEL 1 »(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.
released 2026-09-23 | 3 theorems · 12 lemmas · 18 proofs · 13,187 words |
PLAY LEVEL 2 »(pdf)
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 KG that is injective but not surjective, disproving Gottschalk's surjunctivity conjecture.
released 2026-09-23 | 2 theorems · 5 lemmas · 8 proofs · 3,304 words |
PLAY LEVEL 3 »(pdf)
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 Fugledeā€“Kadison determinant strictly between zero and one. The logarithmic integral defining the determinant is finite.
released 2026-09-26 | 1 theorem · 7 lemmas · 14 proofs · 11,144 words |
PLAY LEVEL 4 »(pdf)
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 KG that is injective but not surjective.