Year: 2026
Type: article
Abstract: A recent preprint [OpenAI, A torsion-free group algebra with zero divisors, preprint (September 23, 2026)] constructs a finitely presented torsion-free group [Formula: see text] with a finite two-dimensional classifying space such that [Formula: see text] contains nonzero zero divisors. We extend this to every odd prime [Formula: see text]: there is a finitely presented torsion-free group [Formula: see text], again with a finite two-dimensional classifying space, such that [Formula: see text] contains nonzero elements [Formula: see text] with [Formula: see text]. Together with the case [Formula: see text], Kaplansky’s zero-divisor conjecture therefore fails over every field of positive characteristic. The new ingredients are a signed version of the parity cancellation, made possible by a bipartite random model on one side; a type design built from the projective plane over [Formula: see text] with [Formula: see text] additional letters, which makes every cross-intersection congruent to 1 modulo [Formula: see text]; and a squared-word decay estimate for this design, proved with an explicit test vector. The geometric part of the argument is taken from the characteristic-two construction. We record a line-by-line check of that part, together with the bookkeeping changes it needs. The characteristic-zero case remains open.
Author: S. V. Ivanov
Institution: University of Illinois Urbana-Champaign
Language: English
Topic: Finite Group Theory Research
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221779193