Engel elements in some fractal groups

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′:stG(1)′|=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|G':{{\mathrm{st}}}_G(1)'|=\infty $$\end{document}, 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′:stG(1)′|=∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|G':{{\mathrm{st}}}_G(1)'|=\infty $$\end{document} and being fractal can be dropped.