First-order model theory and Kaplansky's stable finiteness conjecture

Abstract

Using algebraic geometry methods, the third author proved that the group ring of a surjunctive group with coefficients in a field is always stably finite. In other words, every group satisfying Gottschalk's conjecture also satisfies Kaplansky's stable finiteness conjecture. Here we present an alternative proof of this result based on first-order model theory.