Stable Finiteness of Group Rings in Arbitrary Characteristic
by
Francesc Perera
Universitat Autònoma de Barcelona and Queen's University Belfast
Coauthors: Pere Ara (Universitat Autònoma de Barcelona), Kevin C. O'Meara (University of Canterbury)

In the late 1960's, Kaplansky showed that the group algebra of a group G over a field with characteristic zero is directly finite. Even though different proofs were given shortly after, the general problem of deciding the direct finiteness of K[G] in characteristic p>0 has remained open, and virtually no progress has been made for the last 30 years.

The purpose of the talk is to outline a technique that involves the study of translation rings associated to Cayley graphs of amenable groups, and the Sylvester rank functions one can define on them. As a consequence, we obtain that every group ring D[G] of a free-by- amenable group G over a division ring D of arbitrary characteristic is stably finite, thereby settling the above problem in the positive for this wide class of groups.

Date received: April 5, 2001

Copyright © 2001 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cage-24.