arXiv · 2012.01408
On non-surjective word maps on $\mathrm{PSL}_{2}(\mathbb{F}_{q})$
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 $\mathbb Q$.
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Arindam Biswas, Jyoti Prakash Saha. 2020-12-02. On non-surjective word maps on $\mathrm{PSL}_{2}(\mathbb{F}_{q})$. https://arxiv.org/abs/2012.01408
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