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The Krasner–Kaloujnine universal embedding theorem states that any group extension of a group $H$ by a group $A$ is isomorphic to a subgroup of the regular wreath product $A \operatorname{Wr} H$. When studying finitely generated infinite groups (e.g. in geometric group theory), this statement seems to be of limited use as the regular wreath product $A \operatorname{Wr} H$ is infinitely generated.

I was curious to know under what conditions the regular wreath product $A \operatorname{Wr} H$ could be replaced by the restricted wreath product $A \operatorname{wr} H$, which is finitely generated whenever $A$ and $H$ are. I assume this is not always the case?

In particular, under what conditions is a finitely generated solvable group isomorphic to a subgroup of an iterated restricted wreath product of abelian groups?

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  • $\begingroup$ For a counterexample: the Heisenberg group is an extension of (central) $\mathbf{Z}$ by $\mathbf{Z}^2$ but does not embed into $\mathbf{Z}\wr\mathbf{Z}^2$. $\endgroup$
    – YCor
    Commented Sep 1 at 9:28
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    $\begingroup$ Just sort of a random remark, this theorem actually can be useful in the finitely generated world, since if $H$ is finite (and $A$ is finitely generated) then $A \mathrm{Wr} H$ is finitely generated (sort of cheating since it just equals the restricted version). For example this is the key to proving that if a finite index subgroup of your group embeds into Thompson's group $V$, then your group embeds in $V$. $\endgroup$
    – Matt Zaremsky
    Commented Sep 1 at 12:08

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This is not an answer, but rather a long comment. The point I would like to highlight is that (restricted) wreath products are not finitely presentable, as it is well-known, but also they contain only few finitely presented subgroups. Therefore, if $G$ is a finitely presented group satisfying a short exact sequence $$1 \to A \to G \to H \to 1,$$ typically $G$ will not embed into $A \wr H$. To be more precise, the following statement holds:

Proposition. Let $A,B,G$ be three groups with $G$ finitely presented. Then $G$ is isomorphic to a subgroup of $A \wr B$ if and only if it is isomorphic to a subgroup of $B$ or if there exists some $n \geq 1$ such that $G$ is a (subgroup of $A^n$)-by-(finite subgroup of $B$).

A proof can be found in the paper A note on morphisms to wreath products.

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  • $\begingroup$ Note: the proposition is highly specific to $G$ being finitely presented (otherwise take $A=B=\mathbf{Z}$, $G=\mathbf{Z}\wr\mathbf{Z}$). Suitably reformulated (as in your paper) it indeed implies as a corollary Baumslag's result that wreath products are not fp. $\endgroup$
    – YCor
    Commented Sep 1 at 9:26

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