A torsion-free group algebra that is not directly finite

The explanation

Take an endless list of numbers, like 5, 7, 9, and so on. Let S shift everything right and put 0 in front, giving 0, 5, 7, 9. Let L drop the first entry. Apply S and then L, and you get 5, 7, 9 back. So L undoes S. Now apply L first and then S. You get 0, 7, 9. The 5 is gone. So L is a left inverse of S but not a right inverse.

For an ordinary square matrix this cannot happen. If AB is the identity, then BA is too. The endless list escaped that rule because it has infinitely many entries. There is also a neat certificate. Let c be the list 1, 0, 0, and so on. L sends c to all zeros, but c is not zero. If L followed by S were the identity, then c would equal S applied to L of c, which is zero. So one nonzero c with L c = 0 proves the order matters.

A ring is directly finite when ab = 1 always forces ba = 1. A group is a set of moves that can be combined and undone. A group algebra is made of finite sums of moves, with coefficients from a field, multiplied out like brackets. Each element uses only finitely many moves, so it feels finite. Kaplansky's direct-finiteness conjecture asserts that every such group algebra is directly finiteSource 1.

Over the complex numbers and other fields of characteristic zero, this was proved long ago, by an operator argumentSource 2. Characteristic is about which whole number equals zero in the field: in arithmetic modulo 2, 1 + 1 = 0. In positive characteristic, positive results came for large classes of groups. Elek and Szabó proved it for every sofic group. A sofic group is one whose multiplication can be imitated, ever more closely, by shuffles of finite sets. Whether every group is sofic is itself a well-known open question.

The newest paper claims the conjecture fails even in a very tame settingSource 3. It builds a group with finitely many moves and rules, where no move except doing nothing ever returns to the start. That is called torsion-free. It gives sums a, b and c, counting each move odd or even times, with ab = 1, ac = 0 and c not zeroSource 4. Exactly as with the endless list, c certifies that ba is not 1Source 5. Because Elek and Szabó's theorem would forbid this, the group cannot be soficSource 6.

The cancellation comes from parity. The paper builds labelled graphs and reads group moves off their paths. When certain pieces of a graph have an even number of points, their contributions cancel modulo 2Source 7. Seven extra labels, arranged around a small finite geometry called the Fano plane, leave exactly one piece that does not cancelSource 8. That leftover piece gives the product 1.

The hard part is not the parity count. It is keeping the group honest, so that distinct paths really give distinct moves and no hidden finite-order move appearsSource 9. The paper rules out the bad configurations with a counting estimate and a planar cutting argument.

Three companion papers come at the problem from other sides. One gives a counterexample over a finite field of characteristic 2, for a finitely presented group that has a move of odd prime orderSource 10. Its recipe is proved to finish but has never been runSource 11. It also yields an injective cellular automaton that is not surjective, against Gottschalk's conjectureSource 12. A cellular automaton updates a pattern on the group by one fixed local rule. Another paper handles one huge, specific odd primeSource 13.

The last paper disproves a related claim about determinantsSource 14. An ordinary integer matrix that can be inverted has determinant at least 1 in sizeSource 15. The Fuglede–Kadison determinant extends this idea to matrices of group sums. The paper claims a matrix of integer group sums whose determinant lies strictly between 0 and 1.

How sure is this? A computer proof checker called Lean has checked formal statements for three of the papers. They cover the characteristic-2 counterexample with odd-order moves, the odd-prime counterexample with its cellular automaton, and the determinant claim. Lean does not check the newest torsion-free paper. It also leaves out the conclusion that the groups are not sofic. No specialist review of these papers has been published.

What it doesn't show. This does not write down a group or elements small enough to check by hand.