Words with few values in finite simple groups. (Q2871303)

Let \(G_w=\{w(g_1,\ldots,g_d)^{\pm 1}\mid g_i\in G\}\), where \(w\) is a word in the free group on \(x_1,\ldots,x_d\). In this paper the authors show that for \(n\geq 7\), \(n\neq 13\), there is a word \(w(x_1,x_2)\) such that \(\text{Alt}(n)_w\) consists of the identity and all 3-cycles, for \(n=13\) there is a word \(w(x_1,x_2,x_3)\) with the same property. They also prove a similar result for \(\text{SL}_n(q)\), \((n,q)\neq(4,2)\), and transvections instead of 3-cycles. From this it follows that for any \(k\), there exists a word \(w\) and a finite simple group \(G\), such that \(w\) is not an identity in \(G\) but \(G\neq(G_w)^k\).