Kaplansky's conjectures fail for torsion-free hyperbolic groups in positive characteristic
Do Kaplansky's group-ring conjectures hold for torsion-free word-hyperbolic groups? We show that all four fail in positive characteristic. For every prime $p$ there is a finitely presented, torsion-free, word-hyperbolic group $G$ with a finite two-dimensional classifying space and $a,b\in\mathbb{F}_p[G]$ with $ab=1\neq ba$. So, over every field of characteristic $p$, the group algebra of $G$ is not directly finite and has a non-trivial idempotent, zero divisor and unit, and $G$ is neither sofic nor surjunctive. For $p=2$, $G$, $a$ and $b$ come from OpenAI's random construction, and we add that $G$ is hyperbolic with probability $1-o(1)$, as are OpenAI's zero-divisor groups. For odd $p$ the groups are new: they adapt OpenAI's direct-finiteness construction with a signed cancellation, found first by S. V. Ivanov and independently by us, via an exact solution of a colour-balance problem and a Perron certificate. As far as we know, they are the first torsion-free counterexamples to direct finiteness and to the idempotent conjecture in odd characteristic. Hyperbolicity comes from one deterministic criterion: if a labelled graph is pattern-free, the group obtained by coning off its components is aspherical and has a linear isoperimetric inequality. We prove it by running OpenAI's exclusion of reduced spheres on van Kampen discs, an accounting OpenAI used for another complex. For every prime $p$, the existence of $G$, $a$ and $b$ and the failure of all four conjectures are formalized in Lean 4, including existence forms of the random-graph estimates used. In characteristic zero, where the idempotent and direct-finiteness conjectures hold for torsion-free hyperbolic groups, we show that no cancellation built into the vertex types survives the decay estimate that makes the groups torsion-free. OpenAI's characteristic-two counterexample with torsion is a CAT(0) group.