Some closure results for $\mathcal{C}$-approximable groups

Derek F. Holt and Sarah Rees

Vol. 287 (2017), No. 2, 393–409

DOI: 10.2140/pjm.2017.287.393

Abstract

We investigate closure results for $\mathcal{C}$-approximable groups, for certain classes $\mathcal{C}$, of groups with invariant length functions. In particular we prove, each time for certain (but not necessarily the same) classes $\mathcal{C}$ that: (i) the direct product of two $\mathcal{C}$-approximable groups is $\mathcal{C}$-approximable; (ii) the restricted standard wreath product $G\wr H$ is $\mathcal{C}$-approximable when $G$ is $\mathcal{C}$-approximable and $H$ is residually finite; and (iii) a group $G$ with normal subgroup $N$ is $\mathcal{C}$-approximable when $N$ is $\mathcal{C}$-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 by A.Thom). Our wreath product result is valid for weakly sofic groups, and we prove it separately for sofic groups. This last result has recently been generalised by Hayes and Sale, who proved 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 Szabó (2006) for sofic groups $N$.

Keywords

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

Mathematical Subject Classification 2010

Primary: 20F65

Secondary: 20E22

Milestones

Received: 8 January 2016

Revised: 25 September 2016

Accepted: 26 September 2016

Published: 9 March 2017