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

Abstract

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.

v4: Final version. To appear in Communications in Algebra. A new section was added, which deals with the case when a group has a co-amenable subgroup that is weakly sofic, sofic, linear sofic, or hyperlinear, respectively. 13 pages

References in corpus (3)

• On co-amenability for groups and von Neumann algebras
• Some closure results for C-approximable groups
• Permanence properties of verbal products and verbal wreath products of groups