On Palindromic Widths of Nilpotent and Wreathe Products

TL;DR

This paper investigates the conditions under which the palindromic width of nilpotent group products is finite, and provides new proofs and examples related to the infinite nature of commutator widths in certain wreath products.

Contribution

It establishes a criterion for finite palindromic width in nilpotent products and offers a new proof regarding the infinite commutator width of specific wreath products, expanding understanding of these group properties.

Findings

Nilpotent product has finite palindromic width iff each factor does.

The commutator width of $F_n times K$ is infinite for free group $F_n$ and finite group $K$.

Examples of groups with infinite commutator width but finite palindromic width are provided.

Abstract

We prove that the nilpotent product of a set of groups $A_{1},\dots, A_{s}$ has finite palindromic width if and only if the palindromic widths of $A_{i}, i=1,\dots, s,$ are finite. We give a new proof that the commutator width of $F_n \wr K$ is infinite, where $F_n$ is a free group of rank $n \geq 2$ and $K$ a finite group. This result, combining with a result of Fink \cite{f1} gives examples of groups with infinite commutator width but finite palindromic width with respect to some generating set.