Failure of direct finiteness for group algebras in every odd characteristic

Elena I. Bunina; Winston Lee; Alexei Miasnikov; Kyrylo Muliarchyk

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arXiv · 2610.22536

Failure of direct finiteness for group algebras in every odd characteristic

Elena I. Bunina·Winston Lee·Alexei Miasnikov·Kyrylo Muliarchyk

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Abstract

For every odd prime $p$, we construct a finitely presented group $G_p$ and elements $a,b,c\in\mathbb{F}_p[G_p]$ such that $ab=1$, $ac=0$, and $c\ne0$. Thus $\mathbb{F}_p[G_p]$, 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.

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Elena I. Bunina, Winston Lee, Alexei Miasnikov, Kyrylo Muliarchyk. 2026-10-15. Failure of direct finiteness for group algebras in every odd characteristic. https://arxiv.org/abs/2610.22536

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