On palindromic width of certain extensions and quotients of free nilpotent groups

In arXiv:1303.1129, the authors provided a bound for the palindromic width of free abelian-by-nilpotent group $AN_n$ of rank $n$ and free nilpotent group ${\rm N}_{n,r}$ of rank $n$ and step $r$. In the present paper we study palindromic widths of groups $\widetilde{AN}_n$ and $\widetilde{\rm N}_{n,r}$. We denote by $\widetilde{G}_n = G_n / \langle \langle x_1^2, \ldots, x_n^2 \rangle \rangle$ the quotient of group $G_n = \langle x_1, \ldots, x_n \rangle$, which is free in some variety by the normal subgroup generated by $x_1^2, \ldots, x_n^2$. We prove that the palindromic width of the quotient $\widetilde{AN}_n$ is finite and bounded by $3n$. We also prove that the palindromic width of the quotient $\widetilde{\rm N}_{n, 2}$ is precisely $2(n-1)$. We improve the lower bound of the palindromic width for ${\rm N}_{n, r}$. We prove that the palindromic width of ${\rm N}_{n, r}$, $r\geq 2$ is at least $2(n-1)$. We also improve the bound for palindromic widths of free metabelian groups. We prove that the palindromic width of free metabelian group of rank $n$ is at most $4n-1$.