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.

Keywords:

• Amenable groups
• hyperlinear groups
• linear sofic groups
• sofic groups
• unrestricted wreath products
• weakly sofic groups

2020 Mathematics Subject Classification:

• 20E22
• 20F65
• 43A05

Acknowledgments

We thank Prof. Goulnara Arzhantseva for her comments and for clarifying a historical inaccuracy in the first draft of this article. We thank the referee for his or her careful reading of the manuscript, in particular Section 5 was added thanks to his/her comments.

Additional information

Funding

J. Brude was supported in part by a CONICET Doctoral Fellowship. R. Sasyk was supported in part through the grant PIP-CONICET 11220130100073CO.