Unrestricted wreath products and sofic groups

TL;DR

This paper proves that unrestricted wreath products of sofic groups with amenable groups are sofic, offering an alternative proof for group extensions with sofic kernels and extending results to hyperlinear-by-amenable groups.

Contribution

It introduces a new proof technique for soficity of certain group constructions using the Kaloujnine-Krasner theorem and extends results to hyperlinear-by-amenable groups.

Findings

Unrestricted wreath product of a sofic group by an amenable group is sofic.

Group extensions with sofic kernel and amenable quotient are sofic.

Extension to hyperlinear-by-amenable groups.

Abstract

We show that the unrestricted wreath product of a sofic group by an amenable group is sofic. We use this result to present an alternative proof of the known fact that any group extension with sofic kernel and amenable quotient is again a sofic group. Our approach exploits the famous Kaloujnine-Krasner theorem and extends, with an additional argument, to hyperlinear-by-amenable groups.