Abstract
We consider a family of finitely presented groups, called universal left invertible element (ULIE) groups, that are universal for existence of one-sided invertible elements in a group ring K[G], where K is a field or a division ring. We show that for testing Kaplansky’s direct finiteness conjecture, it suffices to test it on ULIE groups, and we show that there is an infinite family of nonamenable ULIE groups. We consider the invertibles conjecture, and we show that it is equivalent to a question about ULIE groups. By calculating all the ULIE groups over the field of two elements, for ranks (3, n), n ≤ 11, and (5, 5), we show that the direct finiteness conjecture and the invertibles conjecture (which implies the zero divisors conjecture) hold for these ranks over .
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