On equations over groups (Q1061851)

Let G be a group and \(G*<x>\) be the free product of G with the infinite cyclic group \(<x>\). Let \(w(x)=x^{i_ 1}g_ 1x^{i_ 2}g_ 2...x^{i_ k}g_ k\) be an element of \(G*<x>\) such that the total degree of w(x) in x is \(\sum i_ s=d\neq 0\). The authors construct an overgroup of G (i.e. a group L in which G is embedded) such that the equation \(w^ n(x)=1\) has a solution in L for some positive integer n. The proof is too technical to be sketched here.