Some word maps that are non-surjective on infinitely many finite simple groups.
type: Item
instance of: scholarly article
zbMATH Open document ID: 1292.20014
author: Sebastian Jambor
author: Martin W. Liebeck
author: E. A. O'Brien
published in: Bulletin of the London Mathematical Society
publication date: 2013-10-18T00:00:00Z
full work available at URL: https://arxiv.org/abs/1205.1952
Mathematics Subject Classification ID: 20D05
Mathematics Subject Classification ID: 20F10
Mathematics Subject Classification ID: 20F70
zbMATH DE Number: 6216262
zbMATH Keywords: word maps
zbMATH Keywords: finite simple groups
MaRDI profile type: Publication
OpenAlex ID: W3106100766
arXiv ID: 1205.1952
DOI: 10.1112/BLMS/BDT010
title: Some word maps that are non-surjective on infinitely many finite simple groups.
review text: A \textit{word map} is a function \(G^k\to G\), \((x_1,\ldots,x_k)\mapsto w(x_1,\ldots,x_k)\), where \(G\) is a group and \(w\) is a word in \(F_k\), the free group of rank \(k\). The question of which groups \(G\) and which words \(w\) yield surjective word maps, has received much recent attention.NEWLINENEWLINE This paper defines a family of words each of which are non-surjective for an infinite family of groups \(G\cong\mathrm{PSL}_2(q)\), for some \(q\). This is a counterexample to a conjecture of Shalev; moreover, the given words are believed to be the first non-power words (i.e. words not of the form \(x_1^n\)) to be proved non-surjective on an infinite family of finite simple groups.NEWLINENEWLINE The main theorem is as follows: let \(k\geq 2\) be an integer such that \(2k+1\) is prime, and let \(w\) be the word \(x_1^2[x_1^{-2},x_2^{-1}]^k\). Let \(p\neq 2k+1\) be a prime of inertia degree \(m>1\) in \(\mathbb Q(\zeta+\zeta^{-1})\) where \(\zeta\) is a primitive \((2k+1)\)-th root of unity, and \((2/p)=-1\). Then the word map \((x,y)\mapsto w(x,y)\) is non-surjective on \(\mathrm{PSL}_2(q)\) for all \(q=p^n\) where \(n\) is a positive integer divisible neither by \(2\) nor by \(m\).NEWLINENEWLINE To help understand the statement of this theorem, the authors mention the following specific case of the main result: If \(w=x_1^2[x_1^{-2},x_2^{-1}]^2\), then the word map \((x,y)\mapsto w(x,y)\) is non-surjective on \(\mathrm{PSL}_2(p^{2r+1})\) for all non-negative integers \(r\) and all odd primes \(p\neq 5\) such that \(p^2\not\equiv 1\pmod{16}\) and \(p^2\not\equiv 1\pmod 5\).NEWLINENEWLINE The authors prove this result using a classical result of Fricke and Klein that implies that, for every word \(w\) in \(F_2\), there is a unique polynomial \(\tau(s,t,u)\in\mathbb Z[s,t,u]\) such that, for every field \(K\) and for all \(x,y\in\mathrm{SL}_2(K)\), NEWLINE\[NEWLINE\mathrm{Tr}(w(x,y))=\tau(\mathrm{Tr}(x),\mathrm{Tr}(y),\mathrm{Tr}(xy)).NEWLINE\]NEWLINE Here we …
reviewed by: Nick Gill
description: scientific article; zbMATH DE number 6216262