The solution of length four equations over groups

Abstract:

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