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 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.
Classification :
20-XX
Mots-clés :
verbal products of groups, verbal wreath products, sofic groups, hyperlinear groups, linear sofic groups, weakly sofic groups, Haagerup property, Kazhdan’s property (T), exact groups
Affiliations des auteurs :
Javier Brude
1
;
Román Sasyk
2
1
CONICET and Universidad Nacional de la Plata, Buenos Aires, Argentina
2
CONICET and Universidad de Buenos Aires, Argentina
Javier Brude; Román Sasyk. Permanence properties of verbal products and verbal wreath products of groups. Groups, geometry, and dynamics, Tome 16 (2022) no. 2, pp. 363-401. doi: 10.4171/ggd/665
@article{10_4171_ggd_665,
author = {Javier Brude and Rom\'an Sasyk},
title = {Permanence properties of verbal products and verbal wreath products of groups},
journal = {Groups, geometry, and dynamics},
pages = {363--401},
year = {2022},
volume = {16},
number = {2},
doi = {10.4171/ggd/665},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ggd/665/}
}
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AU - Javier Brude
AU - Román Sasyk
TI - Permanence properties of verbal products and verbal wreath products of groups
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