A counterexample to Kaplansky’s zero-divisor conjecture

Constructs a finitely presented torsion-free group G whose group algebra \mathbb F_2[G] has nonzero zero divisors, disproving Kaplansky's zero-divisor conjecture. The group has a finite two-dimensional classifying space. (Lean)

Manuscript

A Torsion-Free Group Algebra with Zero Divisors

September 23, 2026 · source · Lean formalization available · cite as \cite{OAI:A-Torsion-Free-Group-Algebra-with-Zero-Divisors-September-23-2026}

We disprove Kaplansky's zero-divisor conjecture by constructing a finitely presented torsion-free group G for which \mathbb F_2[G] has nonzero zero divisors. The group admits a finite two-dimensional classifying space.

Result 196 · 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.