Sofic actions, halo products, and metric approximations of groups

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 ashalo 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.