Palindromic width of wreath products, metabelian groups, and max-n solvable groups. (Q472160)

Let \(G\) be a group and let \(X\) be a symmetric generating set (that is, \(X=X^{-1}\)). A word in the elements of \(X\) which reads the same forwards as well as backwards is called a `\textit{palindrome}'. If there is a positive integer \(k\) such that every element of \(G\) is a product of at the most \(k\) palindromes, then one says that \(G\) has finite palindromic width with respect to \(X\). The smallest such \(k\) is denoted by \(PW(G,X)\) and is called the palindromic width of \(G\) with respect to \(X\). In a series of papers, Valeriy Bardakov and Krishnendu Gongopadhyay studied this interesting property. In particular, they obtained bounds for the palindromic width of a free \(r\)-step nilpotent group with respect to the standard generating set. They deduced that finitely generated free metabelian groups have finite palindromic width. They have also shown that finitely generated solvable groups satisfying the maximal condition for normal subgroups admit finite generating sets with respect to which they have finite palindromic width. They have also raised a number of interesting questions. In the paper under review, the authors study the palindromic width of wreath products of the form \(G\wr\mathbb Z^r\) in terms of the palindromic width of \(G\). More precisely, they prove: Let \(G\) be a group with finite, symmetric generating set \(X\). Let \(S\) be the standard generating set of \(\mathbb Z^r\). Then, \[ PW(G\wr\mathbb Z^r,X\cup S)\leq 3r+PW(G,X). \] Moreover, if \(r=1\), \(PW(G\wr\mathbb Z,X\cup S)\leq 2+PW(G,X)\). Finally, \(PW(\mathbb Z\wr\mathbb Z)=3\). Their method of proof is to express a finitely supported function from \(\mathbb Z^r\) to a group as a pointwise product of \(r\) such functions which possess some additional symmetry properties. This idea allows the authors here to give new, self-contained proofs of the Bardakov-Gongopadhyay results on finite palindromic widths of free metabelian groups and of finitely generated solvable groups satisfying the maximum condition on normal subgroups. Two interesting general questions which remain open, are: (i) Is it true that a finitely generated group has finite palindromic width if and only if the commutator subgroup has finite width with respect to the set of commutators? (ii) Is it true that a finitely generated group has finite palindromic width with respect to one set of finite generators if and only if it has finite palindromic width with respect to every finite set of generators?