Permanence properties of verbal products and verbal wreath products of groups

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.