A Counterexample to Kaplansky's Direct-Finiteness Conjecture in Characteristic Two
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.
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100+ reactions to 100+ solutionsProofs and PromptsOct 8, 2026
It follows trivially that the latter document provides a counterexample to the idempotent conjecture in characteristic p = 2 p=2 . Indeed, they have a torsion-free group G G and the field with two elements K K , and find elements a,b in the group algebra K [ G ] K[G] satisfying a b = 1 ab=1 and b…