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. introductionGiven two groups G and H, their unrestricted wreath product G ≀≀ H is, by definition, the semidirect product ( H G) ⋊ θ H, where H acts on the direct product H G by shifting coordinates as follows: θ h x h h∈H := x h h h∈H , for x h h∈H ∈ H G. The purpose of this article is to provide a simple and unified proof of the following statement.1.1. Theorem. Let H be an amenable group and let G be a group.(1By means of the Kaloujnine-Krasner theorem, [12], and by keeping in mind that these four metric approximation properties pass to subgroups and are preserved under taking direct products, it is easy to see that Theorem 1.1 and the next result are equivalent. Corollary (Extension Theorem).Let G be a group with a normal subgroup N such that the quotient G/N is amenable. If N is weakly sofic, sofic, linear sofic, or hyperlinear; then G is weakly sofic, sofic, linear sofic, or hyperlinear, respectively.The sofic and hyperlinear cases of Theorem 1.1 have been proved by Arzhantseva, Berlai, and Glebsky in [2], where they also gave the application of the Kaloujnine-Krasner theorem mentioned above. Some of the extension results are older. Indeed, the sofic one is due to Elek and Szabo in [7] and the linear sofic one is due to Arzhantseva and Păunescu in [3]. More recently, in [11], Holt and Rees showed that certain metric approximations on groups, including weakly sofic, are preserved under taking extensions with amenable quotients. In a slightly different direction, in [10], Hayes and Sale proved that the restricted wreath product of a group G having one of the metric approximation properties listed in Theorem 1.1 by an acting sofic group, preserves the approximation property of G.In [2, §4.3] the authors explained why their techniques could not deal with the weakly sofic case of Theorem 1.1. The motivation of the present article was to see whether ideas we used in [4, §5], some of which can be traced back to [10,11], would serve to give a direct proof of this fact, without requiring the result on extensions of [11]. Here we achieve this in a self-contained manner that also allows us to