A geometric generalization of Kaplansky's direct finiteness conjecture

TL;DR

This paper extends Kaplansky's direct finiteness conjecture to a geometric setting involving symbolic algebraic varieties and groups like sofic or surjunctive groups, linking it to surjunctivity.

Contribution

It introduces a geometric generalization of the conjecture for endomorphisms of algebraic varieties and connects stable finiteness to surjunctivity.

Findings

Proves a geometric direct finiteness theorem for certain group rings.

Generalizes Kaplansky's conjecture to a broader class of groups.

Shows stable finiteness follows from surjunctivity.

Abstract

Let $G$ be a group and let $k$ be a field. Kaplansky's direct finiteness conjecture states that every one-sided unit of the group ring $k[G]$ must be a two-sided unit. In this paper, we establish a geometric direct finiteness theorem for endomorphisms of symbolic algebraic varieties. Whenever $G$ is a sofic group or more generally a surjunctive group, our result implies a generalization of Kaplansky's direct finiteness conjecture for the near ring $R(k, G)$ which is $k[X_g\colon g \in G]$ as a group and which contains naturally $k[G]$ as the subring of homogeneous polynomials of degree one. We also prove that Kaplansky's stable finiteness conjecture is a consequence of Gottschalk's Surjunctivity conjecture.