JournalsggdOnline First1 July 2022

Permanence properties of verbal products and verbal wreath products of groups

  • Javier Brude

    CONICET and Universidad Nacional de la Plata, Buenos Aires, Argentina
  • Román Sasyk

    CONICET and Universidad de Buenos Aires, Argentina
Permanence properties of verbal products and verbal wreath products of groups cover
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Abstract

By means of analyzing the notion of verbal products of groups, we show that soficity, hyperlinearity, amenability, the Haagerup property, the Kazhdan property (T), and exactness are preserved under taking kk-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.

Cite this article

Javier Brude, Román Sasyk, Permanence properties of verbal products and verbal wreath products of groups. Groups Geom. Dyn. (2022), published online first

DOI 10.4171/GGD/665