Palindromic Width of Wreath Products

We show that the wreath product $G\wr {\mathbb{Z}}^n$ of any finitely generated group $G$ with ${\mathbb{Z}}^n$ has finite palindromic width. We also show that $C\wr A$ has finite palindromic width if $C$ has finite commutator width and $A$ is a finitely generated infinite abelian group. Further we prove that if $H$ is a non-abelian group with finite palindromic width and $G$ any finitely generated group, then every element of the subgroup $G^{\mathrm{\prime }}\wr H$ can be expressed as a product of uniformly boundedly many palindromes. From this we obtain that $P\wr H$ has finite palindromic width if $P$ is a perfect group and further that $G\wr F$ has finite palindromic width for any finite, non-abelian group $F$.