Abstract
Let p be a prime and let G be a subgroup of a Sylow pro-p subgroup of the group of automorphisms of the p-adic tree. We prove that if G is fractal and \(|G':{{\mathrm{st}}}_G(1)'|=\infty \), then the set L(G) of left Engel elements of G is trivial. This result applies to fractal nonabelian groups with torsion-free abelianization, for example the Basilica group, the Brunner–Sidki–Vieira group, and also to the GGS-group with constant defining vector. We further provide two examples showing that neither of the requirements \(|G':{{\mathrm{st}}}_G(1)'|=\infty \) and being fractal can be dropped.
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We thank Alejandra Garrido and Jone Uria-Albizuri for communicating Lemma 3 to us.
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Communicated by A. Constantin.
All three authors were supported by the Spanish Government Grant MTM2017-86802-P and by the Basque Government Grant IT974-16. The first author was also supported by the Spanish Government Grant MTM2014-53810-C2-2-P, the second author by the ERC Grant 336983, and the third author by the “National Group for Algebraic and Geometric Structures, and their Applications” (GNSAGA - INdAM).
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Fernández-Alcober, G.A., Garreta, A. & Noce, M. Engel elements in some fractal groups. Monatsh Math 189, 651–660 (2019). https://doi.org/10.1007/s00605-018-1218-3
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DOI: https://doi.org/10.1007/s00605-018-1218-3