Zero divisors in group algebras of torsion-free groups in odd characteristic

A recent preprint constructs a finitely presented torsion-free group $G$ with a finite two-dimensional classifying space such that ${\mathbb{F}}_2[G]$ 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 ${\mathbb{F}}_p[G_p]$ contains nonzero elements $\alpha ,\beta$ with $\alpha \beta =0$. 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 ${\mathbb{F}}_{p^3}$ 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.