Sofic groups: graph products and graphs of groups.
type: Item
instance of: scholarly article
title: Sofic groups: graph products and graphs of groups.
zbMATH Open document ID: 1319.20034
author: Laura Ciobanu
author: Sarah Rees
author: Derek F. Holt
published in: Pacific Journal of Mathematics
publication date: 2014-11-14T00:00:00Z
full work available at URL: https://arxiv.org/abs/1212.2739
reviewed by: Do Ngoc Diep
Mathematics Subject Classification ID: 20F65
Mathematics Subject Classification ID: 37B05
Mathematics Subject Classification ID: 20E06
Mathematics Subject Classification ID: 43A07
zbMATH DE Number: 6369628
zbMATH Keywords: sofic groups
zbMATH Keywords: graph products
zbMATH Keywords: graphs of groups
MaRDI profile type: Publication
OpenAlex ID: W3105234032
arXiv ID: 1212.2739
DOI: 10.2140/PJM.2014.271.53
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review text: For a finite set \(A\), let \(S(A)\) be the group of all its permutations. For any positive \(\varepsilon\), two elements \(f_1,f_2\) of \(S(A)\) are \(\varepsilon\)-similar, if they are coincidend \(f_1(a)=f_2(a)\) at most at \(\varepsilon|A|\) points. If \(\varepsilon\geq 1\) the condition is trivial. A group \(G\) is called sofic, if for each \(\varepsilon\in(0,1)\) and any finite set \(F\subseteq G\), \(G\) admits a special \((F,\varepsilon)\)-quasi-action (Definition 1.3), i.e. there is a function \(\varphi\colon G\to S(F)\) with properties: [a] \(\varphi(1)=1\); [b] \(\varphi(g^{-1})=\varphi(g)^{-1}\), \(\forall g\in G\); [c] for any \(g\neq 1\), \(\varphi(g)\) has no fixed points; [d] for any \(g_1,g_2\in F\) the elements \(\varphi(g_1g_2)\) and \(\varphi(g_1)\varphi(g_2)\) are \((F,\varepsilon)\)-similar.NEWLINENEWLINE In the paper under review, the authors prove that: 1. The graph product of sofic groups is sofic (Theorem 1.1 and \S2); 2. The fundamental group (Definition 3.1) of a graph of groups is sofic if each vertex group is sofic and each edge group is amenable (Theorem 1.2 and \S3).
description: scientific article; zbMATH DE number 6369628