Permanence properties of verbal products and verbal wreath products of groups

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