Non-singular Equations over Groups I

Abstract

Let G be a group, t an element distinct from G and r(t)= g1tl1 ⋯ gktlk∈ G ∗ 〈t 〉, where each gi is an element of G of order greater than 2 and the li are non-zero integers such that l1+l2+ ⋯ +lk≠ 0 and |li| ≠ |lj| for i ≠ j. It is known that if k≤ 2, then the natural map from G to the one-relator product 〈G,t | r(t)〉 is injective. In this paper, we prove that the same holds for all k ∉ {4, 5}.