Sofic actions, halo products, and metric approximations of groups
Vadim Alekseev, Henry Bradford
TL;DR
The paper develops a unifying framework for metric approximations of groups via sofic, hyperlinear, linear sofic, weakly sofic, and LEF notions and introduces sof ic actions and halo products. It proves that if $\\Gamma$ and $\\Delta$ belong to a class $\\mathcal{C}$ and $\\beta: \\Gamma \to \\Aut(\\Delta)$ is a sof ic $\\mathcal{C}$-action, then the semidirect product $\\Delta \rtimes_{\\beta} \\Gamma$ remains in $\\mathcal{C}$ (Theorem BigTechThm), and extends this to halo-product constructions (wreath, graph wreath, verbal wreath, symmetric/automorphic enrichments). The authors develop a general framework of product- and wreath-compatibility for metric families (sofic, linear sofic, hyperlinear, weakly sofic, LEF) and apply it to a wide array of examples, unifying prior results (Hayes-Sale, Gao–Elayavalli–Patchell, BrudSasyk) and generating numerous new halo-products with preserved approximation properties. They also provide a parallel theory of approximations for actions (LEF and sof ic actions on sets/graphs and groups) and derive LEF and halo-product consequences, including robust residual finiteness criteria in several cases. Overall, the work supplies broad tools for constructing and verifying metric-approximable groups through semidirect and halo-product operations, with implications for understanding the boundary between soficity and non-soficity in group theory.
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.