Finitely presented groups related to Kaplansky's direct finiteness conjecture

Details Finitely presented groups related to Kaplansky's direct finiteness conjecture Finitely presented groups related to Kaplansky's direct finiteness conjecture

2011-12-08

arxiv.org/abs/1112.1790v2

Mathematics

Rings and Algebras

20 pages. version 2 adds a citation and makes minor changes in exposition

Scientific paper

We introduce a family of finitely presented groups, called Universal Left Invertible Element (or 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/Torsion Conjecture and we show that it is equivalent to a question about ULIE groups. We also show that direct finiteness of K[G x H] for all finite groups H implies stable finiteness of K[G].

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