On non-surjective word maps on PSL ₂(F_q)

Jambor–Liebeck–O’Brien showed that there exist non-proper-power word maps which are not surjective on PSL ₂(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 PSL ₂(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 Q , and on SL ₂(K) where K is a number field of odd degree.