About the metric approximation of Higman's group.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1251.20040
author: Andreas Thom
publication date: 2012-08-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/1005.0823
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 22D25
Mathematics Subject Classification ID: 20F05
Mathematics Subject Classification ID: 46M07
zbMATH DE Number: 6064797
zbMATH Keywords: Higman group
zbMATH Keywords: commutator-contractive invariant length functions
zbMATH Keywords: approximation property
zbMATH Keywords: sofic groups
zbMATH Keywords: group von Neumann algebras
zbMATH Keywords: LEF groups
zbMATH Keywords: finitely presented groups
MaRDI profile type: Publication
OpenAlex ID: W2963622310
Wikidata QID: Q61677682
arXiv ID: 1005.0823
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DOI: 10.1515/JGT.2011.102
title: About the metric approximation of Higman's group.
published in: Journal of Group Theory
review text: Let \(G\) be a group. An `invariant length function' is a map \(l\colon G\to[0,1]\) such that \(l(g)=0\) iff \(g=e\), and \(l(gh)\leq l(g)+l(h)\); \(l(g^{-1})=l(g)\); \(l(hg)=l(gh)\) for all \(g,h\in G\). It is `commutator-contractive' if \(l([g,h])\leq 4l(g)l(h)\) for all \(g,h\in G\).NEWLINENEWLINE Let \(\mathcal C\) be a class of groups with an invariant length function. \(G\) is said to have the `\(\mathcal C\)-approximation property' if for all \(g\in G\setminus\{e\}\) there is \(\delta_g>0\) such that for any finite subset \(F\subset G\) and any \(\varepsilon>0\) there exist a group \(C\in\mathcal C\) and a map \(\varphi\colon G\to C\) such that \(\varphi(e)=e\); \(d(\varphi(gh),\varphi(g)\varphi(h))<\varepsilon\) for all \(g,h\in F\); \(l(\varphi(g))\geq\delta_g\) for all \(g\in F\setminus\{e\}\). E.g. the classes of hyperlinear groups or sofic groups can be obtained using a \(\mathcal C\)-approximation property for an appropriate class \(\mathcal C\).NEWLINENEWLINE Let \(\mathcal F_c\) be the class of finite groups with a commutator-contractive invariant length function. The author shows that Higman's group \(H=\langle a_0,a_1,a_2,a_3\mid a_i=[a_{i+1},a_i],\;i\in\mathbb Z/4\mathbb Z\rangle\) does not have the \(\mathcal F_c\)-approximation property.
reviewed by: Vladimir M. Manuilov
description: scientific article; zbMATH DE number 6064797