A problem of expressibility in some amalgamated products of groups (Q2748394)

Let \(w=w(x_1,\dots,x_n)\) be a word in \(n\) variables. For any group \(G\) let \(S=\{w(g_1,\dots,g_n)^{\pm 1},\;g_j\in G\}\) be the set of all values in \(G\) of \(w\)-words and their inverses, and let \(w(G)\) be the subgroup generated by \(S\). Suppose that \(w(F)\neq F\) for a nonabelian free group \(F\) and \(G=A*_HB\) is the amalgamated free product where \(H\neq A\) and \(B\) is the union of at least three double cosets \(HbH\), then for every positive integer \(m\) there are elements in \(w(G)\) which can not be expressed as a product of fewer than \(m\) elements of \(S\). This generalizes a result of the reviewer [in Proc. Camb. Philos. Soc. 64, 573-584 (1968; Zbl 0159.03001)] which dealt with the case where \(G=A*B\), \(A,B\) not both cyclic of order two.