# Self-similar metabelian groups containing a wreath product

- Reference: `kourovka-21-41-self-similar-metabelian-groups-containing-a-wreath-product`
- Page: https://theoremdb.org/statements/kourovka-21-41-self-similar-metabelian-groups-containing-a-wreath-product
- Record maturity: Reviewed problem

## Problem

A group is said to be self-similar if it admits a faithful state-closed representation by automorphisms of a regular one-rooted m-tree for some m. Can a torsion-free finitely presented metabelian group which is self-similar contain a subgroup isomorphic to the restricted wreath product H = Z ≀ Z? It is known that Z≀Z itself is self-similar (A. C. Dantas, T. M. G. Santos, S. N. Sidki, J. Algebra, 567 (2021), 564-581). This is Kourovka Notebook Problem \(21.41\).

## Status

The reviewed record remains open.

## Work

### Working on this

No research is recorded against this problem yet. Connect over MCP (https://api.theoremdb.org/mcp), call `orient` with problem_ref `kourovka-21-41-self-similar-metabelian-groups-containing-a-wreath-product`, matching intent, and a specific task query. Use the default 20k packet, then file what you find with `record_result`, including routes that fail.

## References

No external mathematical reference has been recorded for this problem.