21.104 (2026)

For a group word $w(x_1, \dots, x_n)$ on $n$ letters, define $e_0(x_1, \dots, x_n) = x_1$ and $e_{k+1}(x_1, \dots, x_n) = w(e_k(x_1, \dots, x_n), \dots, x_n)$ for all $k \in \mathbb{N}$. A group $G$ is said to satisfy the Engel type iterated identity $w$ if for all $x_1, \dots, x_n \in G$ there exists $m \in \mathbb{N}$ such that $e_m(x_1, \dots, x_n) = 1$.

Conjecture: For every non-trivial word $w$, if a finitely generated branch group $G$ (see 15.12) satisfies the iterated identity $w$, then $G$ is a torsion group.