Sofic actions, halo products, and metric approximations of groups

TL;DR

This paper introduces the concept of sofic actions of groups, demonstrates their stability under semidirect products for various classes, and constructs numerous new examples of groups within these classes using halo products.

Contribution

It defines sofic $\

Findings

Classes of groups are closed under semidirect products with sofic actions.

Constructs many new examples of sofic, hyperlinear, linear sofic, and weakly sofic groups.

Unifies and extends existing results on group approximations and embeddings.

Abstract

We introduce the notion of a ``sofic $\mathcal{C}$-action'' of one group on another by automorphisms, for $\mathcal{C}$ a class of groups. We show that if $\mathcal{C}$ is the class of (i) sofic, (ii) hyperlinear, (iii) linear sofic or (iv) weakly sofic groups, then the class $\mathcal{C}$ is closed under taking semidirect products with sofic $\mathcal{C}$-action. We use this to construct a wide variety of new examples of groups in the classes (i)-(iv), many of them arising as ``halo products'' in the sense of Genevois-Tessera. We have a parallel set of results producing new examples of semidirect products which are locally embeddable into finite groups. Our framework also unifies existing results in the literature, due to Hayes-Sale; Brude-Sasyk and Gao-Kunnawalkam Elayavalli-Patchell.