Finitely Presented Groups Related to Kaplansky’s Direct Finiteness Conjecture

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 .

2000 AMS Subject Classification::

• 20C07
• 20E99

Keywords:

• Kaplansky’s direct finiteness conjecture
• Invertibles conjecture
• sofic groups

Notes

¹The code that implements this algorithm and the raw output of it are included in the directory ULIE.computations, which is retrievable from the version of this work available at arXiv1112.1790 as part of the source code.