Kourovka Notebook Problem 21.42

Definitions and mathematical statement

For a group $G$, put $\gamma_1(G)=G$ and $\gamma_{i+1}(G)=[\gamma_i(G),G]$, using subgroups generated by commutators. Nilpotency class exactly three means $\gamma_4(G)=1$ and $\gamma_3(G)\ne1$. The group is torsion-free if no nonidentity element has finite order. Here three-generated means that some list of at most three elements generates $G$ as a group.

For $m\ge2$, the rooted regular $m$-ary tree $T_m$ has as vertices all finite words on an $m$-letter alphabet, with an edge joining a word to each extension by one letter. For $a\in\operatorname{Aut}(T_m)$, its section at $v$ is the rooted automorphism $a|_v$ defined by $a(vw)=a(v)a|_v(w)$. A faithful state-closed representation is an injective homomorphism $\rho:G\to\operatorname{Aut}(T_m)$ whose image contains every section of each of its elements. A group admitting one is called self-similar.

Does there exist a three-generated torsion-free group of nilpotency class three which is not self-similar? A negative self-similarity assertion means that no such faithful state-closed representation exists for any finite $m\ge2$, not merely that one selected representation fails.

The tree action is not required to be transitive on its first level, and intransitive faithful state-closed actions must also be ruled out for a witness. The problem asks for existence within the specified three-generator, class-three range; examples requiring more generators or having higher nilpotency class do not meet it.

Short English statement

Is there a three-generated torsion-free nilpotent group of class three with no faithful state-closed action on any regular rooted tree?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2551 (KOU-21.42), 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 21.42. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.
• S3 A. Berlatto and T. Gentil, On self-similarity of finitely generated torsion-free nilpotent groups, arXiv:2509.16947, Definition and Problem 1. https://arxiv.org/html/2509.16947v1.