Engel elements in some fractal groups

2018/01/14 by Gustavo A. Fernández‐Alcober, Fernández-Alcober, Gustavo A., Albert Garreta +3
Computer Science · Mathematics · #20E08 #20F45 #Cellular Automata and Applications #FOS: Mathematics #Group Theory (math.GR) #Mathematical Dynamics and Fractals #semigroups and automata theory

paper · pdf · doi:10.48550/arxiv.1801.04595

openalex publication_date 2018/01/14 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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':st_G(1)'|=∞, 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':st_G(1)'|=∞ and being fractal can be dropped.

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