Zero divisors in group algebras of torsion-free groups in odd characteristic
Abstract
A recent preprint constructs a finitely presented torsion-free group G with a finite two-dimensional classifying space such that 𝔽2[G] contains nonzero zero divisors. We extend this to every odd prime p: there is a finitely presented torsion-free group Gp, again with a finite two-dimensional classifying space, such that 𝔽p[Gp] contains nonzero elements α,β with αβ=0. Together with the case p=2, 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 𝔽p3 with p additional letters, which makes every cross-intersection congruent to $1$ modulo p; 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. The characteristic-zero case remains open.
Related
- Counterexamples to torsion-freeness in the K-theory of crystallographic group C*-algebras
- Simple Torsion-Free Linear Groups from Global Division Algebras
- Kaplansky's conjectures fail for torsion-free hyperbolic groups in positive characteristic
- Failure of direct finiteness for group algebras in every odd characteristic
- Computable presentations of reduced group C*-algebras of hyperbolic groups
BibTeX
@misc{ivanov2026zero,
title = {{Zero divisors in group algebras of torsion-free groups in odd characteristic}},
author = {S. V. Ivanov},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.17662},
eprint = {2610.17662},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.17662},
}
RIS
TY - UNPB
TI - Zero divisors in group algebras of torsion-free groups in odd characteristic
AU - S. V. Ivanov
PY - 2026
DA - 2026/10/13
PB - arXiv
DO - 10.48550/arxiv.2610.17662
UR - https://arxiv.org/abs/2610.17662
AB - A recent preprint constructs a finitely presented torsion-free group G with a finite two-dimensional classifying space such that 𝔽_2[G] contains nonzero zero divisors. We extend this to every odd prime p: there is a finitely presented torsion-free group G_p, again with a finite two-dimensional classifying space, such that 𝔽_p[G_p] contains nonzero elements α,β with αβ=0. Together with the case p=2, 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 𝔽_p^3 with p additional letters, which makes every cross-intersection congruent to $1$ modulo p; 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. The characteristic-zero case remains open.
ER -