Some word maps that are non-surjective on infinitely many finite simple groups.
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 write \(\mathrm{Tr}(v)\) for the trace of an element \(v\in\mathrm{SL}_2(q)\), and the right hand side denotes the evaluation of the polynomial \(\tau\) at the given values. Now, if we take \(w\) to be one of the words from the main theorem, the authors are able to calculate the associated polynomial \(\tau\), and show that, for an infinite number of finite fields \(K\), the polynomial \(\tau\) has no zeros. Thus no element of \(\mathrm{SL}_2(K)\) of the form \(w(x,y)\) can have trace zero, and the main result is proved.
scientific article; zbMATH DE number 6216262