Kourovka Notebook Problem 21.41

Definitions and mathematical statement

For $m\ge2$, let $T_m$ be the rooted tree of finite words over $\{1,\ldots,m\}$, rooted at the empty word, with edges from $v$ to $vi$. A rooted tree automorphism $a$ has a section $a|_v$ at every vertex, characterized by $a(vw)=a(v)\,a|_v(w)$. A group $G$ is self-similar if it has an injective homomorphism $\rho:G\to\operatorname{Aut}(T_m)$ for some $m$ such that every section of every $\rho(g)$ again belongs to $\rho(G)$. No transitivity of the action, and no finite bound on the number of different sections, is required.

A group is torsion-free if $g^k=1$ with $k\ge1$ implies $g=1$. It is finitely presented if it is given by finitely many generators and finitely many relators. It is metabelian if its commutator subgroup is abelian.

Define the restricted wreath product \[ W=\mathbb Z\wr\mathbb Z =\left(\bigoplus_{j\in\mathbb Z}\mathbb Z\right)\rtimes\mathbb Z, \] where the direct sum consists of finitely supported integer sequences and a generator of the last factor shifts their indices by one. The problem asks whether there exists a torsion-free, finitely presented, metabelian, self-similar group $G$ with an injective homomorphism $W\to G$.

The subgroup embedding need not respect a preassigned tree action or wreath-product decomposition. The self-similar action must be faithful on the full ambient group, not just on the embedded copy of $W$. All four conditions on the ambient group must hold simultaneously.

Short English statement

Can a torsion-free finitely presented metabelian self-similar group contain the restricted wreath product $\mathbb Z\wr\mathbb Z$?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, supplied problems.json, record 2550 (KOU-21.41), 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 21.41. Expanded formulation of the supplied catalogue statement. https://arxiv.org/pdf/1401.0300v47.