Permanence properties of verbal products and verbal wreath products of groups

TL;DR

This paper investigates how various group properties like soficity, hyperlinearity, and amenability are preserved under different verbal and wreath product constructions, providing new insights into their structural behavior.

Contribution

It establishes the preservation of key properties under nilpotent, solvable, and Burnside verbal products and wreath products, extending understanding of group property stability.

Findings

Soficity and the Haagerup property are preserved under specific verbal wreath products.

Certain quotients of free groups are shown to be sofic or have the Haagerup property.

Several properties are not preserved under all verbal products, such as orderability.

Abstract

By means of analyzing the notion of verbal products of groups, we show that soficity, hyperlinearity, amenability, the Haagerup property, the Kazhdan's property (T) and exactness are preserved under taking $k$-nilpotent products of groups, while being orderable is not preserved. We also study these properties for solvable and for Burnside products of groups. We then show that if two discrete groups are sofic, or have the Haagerup property, their restricted verbal wreath product arising from nilpotent, solvable and certain Burnside products is also sofic or has the Haagerup property respectively. We also prove related results for hyperlinear, linear sofic and weakly sofic approximations. Finally, we give applications combining our work with the Shmelkin embedding to show that certain quotients of free groups are sofic or have the Haagerup property.