Kaplansky's direct finiteness conjecture

Theorem 1.1 (September 23): a finite field $K$ of characteristic two, a finitely presented group $G$ containing odd-order torsion, and $a,b\in K[G]$ with $ab=1$, $ba\ne1$. By Elek-Szabo, $G$ is not sofic. Companions: for a specified odd prime $p$ (the least prime factor of $(\binom{1200}{600}!)^2+1$), a field of order $p^4$ and a finitely generated group with torsion (September 26); and a finitely presented torsion-free group with a finite two-dimensional classifying complex and $a,b,c\in\mathbb F_2[G]$ with $ab=1$, $ac=0$, $c\ne0$ (October 4). No counterexample in characteristic zero is possible, and none is claimed.

Theorem 1.1 (September 23): a finite field $K$ of characteristic two, a finitely presented group $G$ containing odd-order torsion, and $a,b\in K[G]$ with $ab=1$, $ba\ne1$. By Elek-Szabo, $G$ is not sofic. Companions: for a specified odd prime $p$ (the least prime factor of $(\binom{1200}{600}!)^2+1$), a field of order $p^4$ and a finitely generated group with torsion (September 26); and a finitely presented torsion-free group with a finite two-dimensional classifying complex and $a,b,c\in\mathbb F_2[G]$ with $ab=1$, $ac=0$, $c\ne0$ (October 4). No counterexample in characteristic zero is possible, and none is claimed.