Kourovka Notebook Problem 21.104

Definitions and mathematical statement

Let $T$ be an infinite rooted tree in which vertices at level $j$ have the same finite number $d_j\ge2$ of children. A group $G\le\operatorname{Aut}(T)$ acts faithfully by root-preserving automorphisms. For a vertex $v$, its rigid stabilizer $\operatorname{rist}_G(v)$ consists of elements acting trivially outside the descendant subtree rooted at $v$. Let $\operatorname{rist}_G(j)$ be the subgroup generated by the rigid stabilizers of all vertices at level $j$. The group is a branch group if it is transitive on each level and $[G:\operatorname{rist}_G(j)]<\infty$ for every $j$.

Fix a nontrivial word $w\in F_r$ in $r\ge1$ variables, where nontrivial means nonidentity in the free group. Define word iterates by \[ e_0(x_1,\ldots,x_r)=x_1,\qquad e_{k+1}(x_1,\ldots,x_r) =w(e_k(x_1,\ldots,x_r),x_2,\ldots,x_r). \] The group satisfies this Engel-type iterated identity if, for each tuple $(g_1,\ldots,g_r)\in G^r$, some integer $m\ge1$ has $e_m(g_1,\ldots,g_r)=1$. The iteration count may depend on the tuple; no common bound is assumed.

The conjecture states that a finitely generated branch group satisfying such an iterated identity must be torsion. Thus each $g\in G$ must satisfy $g^n=1$ for some positive integer $n$ depending on $g$. The assertion quantifies over every nontrivial word, not just the usual commutator word, and asks for finite element orders rather than a uniform group exponent.

Short English statement

Must a finitely generated branch group satisfying any nontrivial Engel-type iterated word identity have every element of finite order?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2613 (KOU-21.104), Kourovka Notebook - New Problems, Issue 21. Catalogue identity and source transcription; CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 The Kourovka Notebook, 21st edition, new problems, Problem 21.104. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.