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

Abstract

In arXiv:1303.1129, the authors provided a bound for the palindromic width of free abelian-by-nilpotent group $A{N}_n$ of rank $n$ and free nilpotent group ${\mathrm{N}}_{n,r}$ of rank $n$ and step $r$. In the present paper we study palindromic widths of groups ${\widetilde{AN}}_n$ and ${\tilde{\mathrm{N}}}_{n,r}$. We denote by ${\tilde{G}}_n=G_n/⟨⟨x_1^2,\dots ,x_n^2⟩⟩$ the quotient of group $G_n=⟨x_1,\dots ,x_n⟩$, which is free in some variety by the normal subgroup generated by $x_1^2,\dots ,x_n^2$. 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 ${\tilde{\mathrm{N}}}_{n,2}$ is precisely $2(n-1)$. We improve the lower bound of the palindromic width for ${\mathrm{N}}_{n,r}$. We prove that the palindromic width of ${\mathrm{N}}_{n,r}$, $r\ge 2$ 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 $4n-1$.