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

TL;DR

This paper offers a new proof using first-order model theory that surjunctive groups with coefficients in a field have stably finite group rings, confirming Kaplansky's stable finiteness conjecture for these groups.

Contribution

It provides an alternative proof of a known result by employing first-order model theory instead of algebraic geometry methods.

Findings

Surjunctive groups with coefficients in a field have stably finite group rings

The result confirms Kaplansky's stable finiteness conjecture for these groups

First-order model theory can be effectively used in this context

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.