Automorphic word maps and the Amit-Ashurst conjecture (Q6611931)

Let \(G\) be a finite group and \(F_{k}=\langle x_{1}, x_{2}, \ldots,x_{k} \rangle\) the free group of rank \(k\). The evaluation of an element \(w\in F_{k}\) on \(k\)-tuples \((g_{1}, g_{2}, \ldots, g_{k}) \in G^{k}\) induces a map \(w: G^{k} \rightarrow G\) called the word map on \(G\) induced by \(w\). The image of a word map \(w\) is denoted by \(w(G)\) and, for \(g\in G\), the fiber of \(w\) at \(g\) is the subset \(w^{-1}(g) \subseteq G^{k}\). Let \(P_{w,G}(g)=|w^{-1}(g)|/|G|^{k}\), then the map \(P_{w,G}(g): G \rightarrow [0,1]\) is a probability function on \(G\).\N\NThe authors refer to the Amit-Ashurst conjecture as the statement that \(P_{w,G}(g) \geq |G|^{-1}\) for all finite nilpotent groups \(G\), \(w \in F_{k}\) and \(g \in w(G)\). Toward this conjecture, it was shown by \textit{R. D. Camina} et al. [Arch. Math. 115, No. 6, 599--609 (2020; Zbl 1508.20036)] that if \(p\) is an odd prime and \(G\) is a finite \(p\)-group of nilpotency class 2, then for each \(w \in F_{k}\) and \(g\in w(G)\), the bound \(P_{w,G}(g) \geq |G|^{-2}\) holds.\N\NThe main result in the paper under review is Theorem A: Let \(G\) be a finite nilpotent group of class \(2\) and \(w \in F_{k}\). Then, for each \(g \in w(G)\), we have \(P_{w,G}(g) \geq \big (|G'||G| \big)^{-1}\).\N\NThe paper contains other interesting results in a similar vein. The statements are too complicated to be reported here.