On non-surjective word maps on $\mathrm{PSL}_{2}(\mathbb{F}_{q})$

TL;DR

This paper constructs new examples of non-surjective word maps on (\u210b_q) and applies these findings to the absolute Galois group of , challenging previous conjectures about surjectivity.

Contribution

It provides new explicit examples of non-surjective word maps on (q) and explores their implications for the absolute Galois group of .

Findings

Existence of non-proper-power word maps not surjective on (q) for infinitely many q

Counterexamples to Shalev's conjecture on surjectivity of word maps

Application to non-surjective word maps on the absolute Galois group of 

Abstract

Jambor--Liebeck--O'Brien showed that there exist non-proper-power word maps which are not surjective on $\mathrm{PSL}_{2}(\mathbb{F}_{q})$ for infinitely many $q$. This provided the first counterexamples to a conjecture of Shalev which stated that if a two-variable word is not a proper power of a non-trivial word, then the corresponding word map is surjective on $\mathrm{PSL}_2(\mathbb{F}_{q})$ for all sufficiently large $q$. Motivated by their work, we construct new examples of these types of non-surjective word maps. As an application, we obtain non-surjective word maps on the absolute Galois group of $\mathbb Q$.