Some closure results for C-approximable groups

TL;DR

This paper explores how certain closure properties, like direct products and wreath products, hold for classes of C-approximable groups, extending known results to broader classes such as LEF, sofic, and hyperlinear groups.

Contribution

It establishes new closure results for C-approximable groups under direct products, wreath products, and extensions, broadening the understanding of these groups' structural properties.

Findings

Direct product of two C-approximable groups is C-approximable for several classes.

Wreath product of a C-approximable group with a residually finite group is C-approximable.

Extension of a C-approximable group by an amenable group remains C-approximable.

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 (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. We note that this last result has recently been generalised by Hayes and Sale, who prove that the restricted standard wreath product of any two sofic groups is sofic. Our result on extensions by amenable groups is valid for weakly sofic groups, and was proved by Elek and Szabo for sofic groups N.