Failure of direct finiteness for group algebras in every odd characteristic

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 $c\ne 0$. 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.