Abstract
Jambor–Liebeck–O’Brien showed that there exist non-proper-power word maps which are not surjective on \(\textrm{PSL}_{2}(\mathbb {F}_{q})\) 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 \(\textrm{PSL}_2(\mathbb {F}_{q})\) 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 \(\mathbb Q\), and on \({\text {SL}}_2(K)\) where K is a number field of odd degree.
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Acknowledgements
We thank the anonymous reviewers for their helpful comments and suggestions. The first author wishes to thank Chen Meiri for a number of discussions on word maps. The work of the first author was supported by the ISF Grant no. 1226/19 at the Department of Mathematics at the Technion. The second author acknowledges the Initiation Grant from the Indian Institute of Science Education and Research Bhopal, and the INSPIRE Faculty Award IFA18-MA123 from the Department of Science and Technology, Government of India.
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Biswas, A., Saha, J.P. On non-surjective word maps on \(\textrm{PSL}_{2}(\mathbb {F}_{q})\). Arch. Math. 122, 1–11 (2024). https://doi.org/10.1007/s00013-023-01917-3
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DOI: https://doi.org/10.1007/s00013-023-01917-3