## Some closure results for $\C$-approximable groups

### Derek F. Holt and Sarah Rees

#### Keywords

C-approximable group, sofic, hyperlinear, weakly sofic, linearly sofic

#### Status

To appear in Pac. J. Math..

#### Abstract

We investigate closure results for $\C$-approximable groups,
for certain classes $\C$ of groups with invariant length functions.
In particular we prove, each time for certain (but not necessarily the same)
classes $\C$ that: \linebreak
(i) the direct product of two $\C$-approximable groups is
$\C$-approximable;
(ii) the restricted standard wreath product $G \wr H$ is $\C$-approximable
when $G$ is $\C$-approximable and $H$ is residually finite; and
(iii) a group $G$ with normal subgroup $N$ is $\C$-approximable
when $N$ is $\C$-approximable and $G/N$ is amenable.
Our direct product result is valid for LEF, weakly sofic and hyperlinear
groups, as well as for all groups that are approximable by finite
groups equipped with commutator-contractive invariant length functions
(considered in \cite{Thom}). Our wreath product result is valid for weakly
sofic groups, and we prove it separately for sofic groups. Our result on
extensions by amenable groups is valid for weakly sofic groups, and was proved
in \cite[Theorem 1 (3)]{ElekSzabo} for sofic groups $N$.
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The preprint is available as a gzipped
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