Metric approximations of unrestricted wreath products when the acting group is amenable

Abstract

We give a simple and unified proof that the unrestricted wreath product of a weak sofic, sofic, linear sofic or hyperlinear group by an amenable group is weak 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.