Soficity and other permanence properties of verbal products of groups
Javier Brude, Román Sasyk · arXiv (Cornell University) · 2019
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. We also study these properties for solvable products of groups. We then show that if two discrete groups are sofic, (respectively have the Haagerup property), their restricted verbal wreath product arising from nilpotent or solvable products are also sofic (resp. has the Haagerup Property). Finally, we prove a related result for hyperlinearity.