Linear sofic groups and algebras

TL;DR

This paper introduces linear sofic groups and algebras, exploring their properties, relationships with other classes, and implications for Kaplansky's conjecture, thereby extending the framework of soficity.

Contribution

It systematically defines and studies linear sofic groups and algebras, establishing their properties, relationships with sofic and weak sofic groups, and providing new proofs for existing results.

Findings

Linear sofic groups are equivalent to their group algebras being linear sofic.

Linear soficity is weaker than soficity but stronger than weak soficity.

Sofic groups satisfy Kaplansky's direct finiteness conjecture.

Abstract

We introduce and systematically study linear sofic groups and linear sofic algebras. This generalizes amenable and LEF groups and algebras. We prove that a group is linear sofic if and only if its group algebra is linear sofic. We show that linear soficity for groups is a priori weaker than soficity but stronger than weak soficity. We also provide an alternative proof of a result of Elek and Szabo which states that sofic groups satisfy Kaplansky's direct finiteness conjecture.