Finitely presented groups related to Kaplansky's Direct Finiteness Conjecture
Abstract
We consider 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 non-amenable ULIE groups. We consider the Invertibles Conjecture and we show that it is equivalent to a question about ULIE groups. We also show that for any group G, direct finiteness of K[ G x H ] for all finite groups H implies stable finiteness of K[G]. Thus, truth of the Direct Finiteness Conjecture implies stable finiteness. By calculating all the ULIE groups over the field K=F_2 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 F_2.
Keywords
Cite
@article{arxiv.1112.1790,
title = {Finitely presented groups related to Kaplansky's Direct Finiteness Conjecture},
author = {Ken Dykema and Timo Heister and Kate Juschenko},
journal= {arXiv preprint arXiv:1112.1790},
year = {2015}
}
Comments
44 pages. Version 2 adds a citation and makes minor changes in exposition. Version 3 adds a co-author and the results of computations. Code and raw data associated with the computations have been uploaded with this arXiv submission in the directory ULIE.computations. Retrieve the source code and look in the .tar file for this. (Version 4 is to correct an error in the attachment of this data.)