We study properties of Engel elements in weakly branch groups, lying in the group of automorphisms of a spherically homogeneous rooted tree. More precisely, we prove that the set of bounded left Engel elements is always trivial in weakly branch groups. In the case of branch groups, the existence of non-trivial left Engel elements implies that these are all $p$-elements and that the group is virtually a $p$-group (and so periodic) for some prime $p$. We also show that the set of right Engel elements of a weakly branch group is trivial under a relatively mild condition. Also, we apply these results to well-known families of weakly branch groups, like the multi-GGS groups.

中文翻译：

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弱支化群中的恩格尔元素

我们研究弱分支群中恩格尔元素的性质，这些弱分支群位于球状同质有根树的自同构群中。更准确地说，我们证明了有界左恩格尔元素的集合在弱分支群中总是微不足道的。在分支群的情况下，非平凡左恩格尔元素的存在意味着这些都是 $p$-元素，并且对于某些素数 $p$，该群实际上是一个 $p$-群（因此是周期性的）。我们还表明，在相对温和的条件下，弱分支群的右恩格尔元素集是微不足道的。此外，我们将这些结果应用于众所周知的弱分支群家族，如多 GGS 群。