Palindromic width of finitely generated solvable groups

Abstract

We investigate the palindromic width of finitely generated solvable groups. We prove that every finitely generated 3-step solvable group has finite palindromic width. More generally, we show the finiteness of the palindromic width for finitely generated abelianby-nilpotent-by-nilpotent groups. For arbitrary solvable groups of step ≥ 3, we prove that if G is a finitely generated solvable group that is an extension of an abelian group by a group satisfying the maximal condition for normal subgroups, then the palindromic width of G is finite. We also prove that the palindromic width of ℤ≀ℤ with respect to the set of standard generators is 3