Kaplansky's direct-finiteness conjecture
Does ab = 1 imply ba = 1 in every group algebra? Kaplansky asked this, and the answer is yes in characteristic zero and for sofic groups. OpenAI showed in 2026 that it is no in general: over a finite field of characteristic two there are a finitely presented group G and a, b with ab = 1 and ba != 1. The group is not sofic and carries an injective cellular automaton that is not surjective, disproving Gottschalk's conjecture. The course proves Kaplansky's theorem in characteristic zero and the Gromov-Weiss theorem, then the construction: copies of F_l^2 glued along a system of affine lines of incidence girth 12, character packing, low-degree interpolation, and a line system built from holes, random slabs and the local lemma. The course is an exposition of OpenAI's proof (2026).
Nine lessons with proofs, examples and exercises with solutions. Written and self-checked by Claude Opus 5.5 (Anthropic); independent review is not complete.
- Directly finite group algebras
- Sofic groups and surjunctivity
- The glued group and planar balance
- Subgroup averages and character twists
- Packing and the reverse defect
- Polynomial pairings on punctured spaces
- Direction spaces and compatible holes
- Random slabs and the local lemma
- Line partitions, girth and the counterexample
Theorems and assumptions · Reader and editable sources · Course record
Prerequisites
- Idempotents and integral traces (The Bass trace conjecture for group rings)
- The Kadison–Kaplansky conjecture and scalar compression (A torsion-free counterexample to the Kadison–Kaplansky conjecture)
- The regular trace and spectral phase (The Kervaire conjecture)
- Group algebras and zero divisors (A torsion-free group algebra with zero divisors)
- Groups, presentations and group algebras (recalled in the first lessons).
- Finite fields and linear algebra over them (core courses Abstract Algebra and Linear Algebra).
- Basic discrete probability: expectation, variance, Chebyshev's inequality (core course Probability).