Finitely presented groups related to Kaplansky's direct finiteness conjecture

Ken Dykema*, Timo Heister, Kate Juschenko

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

4 Scopus citations

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 Kaplanskys 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 K= f2of 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 f2.

Original languageEnglish (US)
Pages (from-to)326-338
Number of pages13
JournalExperimental Mathematics
Volume24
Issue number3
DOIs
StatePublished - Jul 3 2015

Keywords

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

ASJC Scopus subject areas

  • Mathematics(all)

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