A torsion-free group algebra with zero divisors

There is a finitely presented torsion-free group G, with a finite two-dimensional classifying space, whose group algebra over the field with two elements has zero divisors: Kaplansky's zero-divisor conjecture fails. Two random immersed labelled graphs with prescribed letter sets of odd intersections, built from the projective plane over the field with 128 elements, make every contribution to a product cancel in pairs; a word-counting estimate, planar separation and cone pictures show that the group of the graphs is torsion-free and that the factors are nonzero. The course is an exposition of OpenAI's proof (2026).

Ten lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.

1. Group algebras and zero divisors
2. Projective planes and prescribed types
3. Random matchings with large girth
4. Image graphs and multiplicity stages
5. Aligned blocks and word counts
6. The bounded-pattern estimate
7. Spherical arrangements and planar intervals
8. Planar separation and the absence of arrangements
9. Cone pictures and asphericity
10. Torsion-freeness and the zero divisors

Theorems and assumptions · Reader and editable sources · Course record

Prerequisites

• Explicit CW models and the first Hurewicz map (Complex threefolds from marked torus families)
• Homotopy fibres and the Serre spectral sequence · Characteristic classes
• Traces of projective modules (The Bass trace conjecture for group rings)
• Cell complexes and homotopy equivalences (Cell complexes, fundamental groups and covering spaces)
• Fundamental groups of graphs and two-complexes (Cell complexes, fundamental groups and covering spaces)
• Universal covers and cellular chains (Cell complexes, fundamental groups and covering spaces)
• The Kadison–Kaplansky conjecture and scalar compression (A torsion-free counterexample to the Kadison–Kaplansky conjecture)
• Directly finite group algebras (Kaplansky's direct-finiteness conjecture)
• Fields with 2^k elements and vector spaces over them (core course Abstract Algebra II).
• Fundamental groups, covering spaces and singular homology (core course Algebraic Topology), the Hurewicz theorem and cellular homology from earlier programme lessons.
• Finite probability: expectations, Markov's inequality and the union bound.