The solution of length four equations over groups

Abstract

Let G G be a group, F F the free group generated by t t and let r ( t ) ∈ G ∗ F r(t) \in G \ast F . The equation r ( t ) = 1 r(t) = 1 is said to have a solution over G G if it has a solution in some group that contains G G . This is equivalent to saying that the natural map G → ⟨ G ∗ F | r ( t ) ⟩ G \to \langle G \ast F|r(t)\rangle is injective. There is a conjecture (attributed to M. Kervaire and F. Laudenbach) that injectivity fails only if the exponent sum of t t in r ( t ) r(t) is zero. In this paper we verify this conjecture in the case when the sum of the absolute values of the exponent of t t in r ( t ) r(t) is equal to four.