Finitely presented groups related to Kaplansky's Direct Finiteness Conjecture

TL;DR

This paper introduces ULIE groups, a universal class for testing Kaplansky's Direct Finiteness Conjecture, and provides evidence supporting the conjecture for specific cases over the field F_2.

Contribution

It defines ULIE groups as a key tool for verifying Kaplansky's conjecture and demonstrates their properties, including non-amenability and their relation to stable finiteness.

Findings

ULIE groups are sufficient for testing the conjecture.

An infinite family of non-amenable ULIE groups is constructed.

The conjecture holds for certain ranks over F_2.

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.