Nonsingular Equations over Groups II

Abstract

Let G be a group, t an element distinct from G, and r(t) = g 1 t l 1 …g k t l k ∈ G* ⟨t⟩, where each g i is an element of G order greater than 2, and the l i are nonzero integers such that l 1 + l 2+…+l k ≠ 0 and |l i | ≠ |l j | for i ≠ j. We prove that if k = 4, then the natural map from G to the one-relator product ⟨G*t | r(t)⟩ is injective. This together with previous results show that the natural map from G is injective for k ≥ 1.