On non-surjective word maps on ${\mathrm{P}\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_q)$

Jambor–Liebeck–O’Brien showed that there exist non-proper-power word maps which are not surjective on ${\mathrm{P}\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_q)$ 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 ${\mathrm{P}\mathrm{S}\mathrm{L}}_2({\mathbb{F}}_q)$ 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 $\mathbb{Q}$.