Kaplansky’s direct finiteness conjecture

Not too long ago I blogged about the first counter-example to Kaplansky’s unit conjecture (link) stating that there are no non-trivial units in the group ring K[G] for K a field and G a torsion-free group. A related conjecture of Kaplansky (one that I was not aware of until recently) is that K[G] is directly finite for K a field and G an arbitrary group. Recall that a ring is called directly finite if every one-sided unit is actually a two-sided one. There are two important results about this direct finiteness conjecture:

  • Kaplansky himself proved the conjecture (for arbitrary groups) for any field K of zero characteristic (Kaplansky: Fields and Rings, Chicago Lectures in Mathematics, 1969).
  • Elek and Szabo proved the conjecture (for arbitrary fields) if the group is sofic (arXiv:math/0305440).

Note that the class of sofic groups is very large. Two important subclasses that it contains are the amenable and the residually finite groups. Further, as far as I know there is currently no example of a group which is provably non-sofic

An even larger class of groups than the sofic ones are surjunctive groups (i.e., every sofic group is surjunctive, https://doi.org/10.1007/PL00011162). In a recent preprint (arXiv:2111.07930) the result of Elek and Szabo was generalized to this larger class of groups.

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