On nonsurjective word maps on $\mathrm{PSL}_{2}(\mathbb{F}_{q})$
Abstract
JamborLiebeckO'Brien showed that there exist nonproperpower 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 twovariable word is not a proper power of a nontrivial 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 nonsurjective word maps. As an application, we obtain nonsurjective word maps on the absolute Galois group of $\mathbb Q$.
 Publication:

arXiv eprints
 Pub Date:
 December 2020
 arXiv:
 arXiv:2012.01408
 Bibcode:
 2020arXiv201201408B
 Keywords:

 Mathematics  Group Theory;
 Mathematics  Number Theory;
 20D05;
 16R30