Palindromic widths of nilpotent and wreath products

Abstract

We prove that the nilpotent product of a set of groups A 1,…,A s has finite palindromic width if and only if the palindromic widths of A i ,i=1,…,s,are finite. We give a new proof that the commutator width of F n ≀K is infinite, where F n is a free group of rank n≥2 and K is a finite group. This result, combining with a result of Fink [9] gives examples of groups with infinite commutator width but finite palindromic width with respect to some generating set.