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

Abstract

In [Bardakov and Gongopadhyay, Palindromic width of free nilpotent groups, J. Algebra 402 (2014) 379-391] the authors provided a bound for the palindromic widths of free abelian-by-nilpotent group ANn of rank n and free nilpotent group Nn,r of rank n and step r. In the present paper, we study palindromic widths of groups &widetilde; ANnand &widetilde; Nn, r. We denote by &widetilde; Gn = Gn/⟨ ⟨ x12,.,xn2 ⟩ ⟩ the quotient of the group Gn = (x1,., xn), which is free in some variety by the normal subgroup generated by x12,.,xn2. We prove that the palindromic width of the quotient &widetilde; ANn is finite and bounded by 3n. We also prove that the palindromic width of the quotient &widetilde;Nn,2 is precisely 2(n-1). As a corollary to this result, we improve the lower bound of the palindromic width of Nn,r. We also improve the bound of the palindromic width of a free metabelian group. We prove that the palindromic width of a free metabelian group of rank n is at most 4n-1.