PreprintCC BY 4.0arXiv
Failure of direct finiteness for group algebras in every odd characteristic
Abstract
For every odd prime p, we construct a finitely presented group Gp and elements a,b,c∈𝔽p[Gp] such that ab=1, ac=0, and c≠0. Thus 𝔽p[Gp], and the group algebra over every field of characteristic p, fails to be directly finite. The construction combines a single exceptional self-product pair with signed bipartite cancellation. New integer matching data realize the exceptional self-intersection while retaining a strict squared-word decay estimate. We verify the adaptations needed for the earlier bounded-path and planar-localization arguments. A minimal disk picture protects the identity coefficient of c, so the proof does not require torsion-freeness.
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BibTeX
@misc{bunina2026failure,
title = {{Failure of direct finiteness for group algebras in every odd characteristic}},
author = {Elena I. Bunina and Winston Lee and Alexei Miasnikov and Kyrylo Muliarchyk},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.22536},
eprint = {2610.22536},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.22536},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Failure of direct finiteness for group algebras in every odd characteristic
AU - Elena I. Bunina
AU - Winston Lee
AU - Alexei Miasnikov
AU - Kyrylo Muliarchyk
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.22536
UR - https://arxiv.org/abs/2610.22536
AB - For every odd prime p, we construct a finitely presented group G_p and elements a,b,c∈𝔽_p[G_p] such that ab=1, ac=0, and c≠0. Thus 𝔽_p[G_p], and the group algebra over every field of characteristic p, fails to be directly finite. The construction combines a single exceptional self-product pair with signed bipartite cancellation. New integer matching data realize the exceptional self-intersection while retaining a strict squared-word decay estimate. We verify the adaptations needed for the earlier bounded-path and planar-localization arguments. A minimal disk picture protects the identity coefficient of c, so the proof does not require torsion-freeness.
ER -