Palindromic width of wreath products

A palindrome is a word which reads the same left-to-right as right-to-left. We show that the wreath product G≀ZnG≀Zn of any finitely generated group G   with ZnZn has finite palindromic width. This generalizes the main result from [16]. We also show that C≀AC≀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′≀HG′≀H can be expressed as a product of uniformly boundedly many palindromes. From this we obtain that P≀HP≀H has finite palindromic width if P   is a perfect group and further that G≀FG≀F has finite palindromic width for any finite, non-abelian group F.