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

الأصل · EN

In arXiv:1303.1129, the authors provided a bound for the palindromic width 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 ANₙ and Nₙ,ᵣ. We denote by Gₙ = Gₙ / x₁²,, xₙ² the quotient of 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 ANₙ is finite and bounded by 3n. We also prove that the palindromic width of the quotient Nₙ, ₂ is precisely 2(n-1). We improve the lower bound of the palindromic width for Nₙ, ᵣ. We prove that the palindromic width of Nₙ, ᵣ, r≥ 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.