Holt, Derek F. and Rees, Sarah (2017) Some closure results for C-approximable groups. Pacific Journal of Mathematics, 287 (2). pp. 393-409. doi:10.2140/pjm.2017.287.393 ISSN 0030-8730.

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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 $N$. \

Item Type:	Journal Article
Subjects:	Q Science > QA Mathematics
Divisions:	Faculty of Science, Engineering and Medicine > Science > Mathematics
Library of Congress Subject Headings (LCSH):	Group theory
Journal or Publication Title:	Pacific Journal of Mathematics
Publisher:	Mathematical Sciences Publishers
ISSN:	0030-8730
Official Date:	9 March 2017
Dates:	Date Event 9 March 2017 Published 26 September 2016 Accepted
Volume:	287
Number:	2
Page Range:	pp. 393-409
DOI:	10.2140/pjm.2017.287.393
Status:	Peer Reviewed
Publication Status:	Published
Access rights to Published version:	Restricted or Subscription Access
Date of first compliant deposit:	20 October 2016
Date of first compliant Open Access:	20 October 2016

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