3 research outputs found
Metric approximations of unrestricted wreath products when the acting group is amenable
We give a simple and unified proof showing that the unrestricted wreath product of a weakly sofic, sofic, linear sofic, or hyperlinear group by an amenable group is weakly sofic, sofic, linear sofic, or hyperlinear, respectively. By means of the Kaloujnine-Krasner theorem, this implies that group extensions with amenable quotients preserve the four aforementioned metric approximation properties. We also discuss the case of co-amenable groups.Fil: Brude, Javier Eugenio. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Sasyk, Roman. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones CientÃficas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin
Permanence properties of verbal products and verbal wreath products of groups
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 -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.Comment: v5: Corrects some typos. No new results added. Final version, to
appear in Groups, Geometry, and Dynamic