Engel elements in some fractal groups

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.