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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

The solution of length four equations over groups


Authors: Martin Edjvet and James Howie
Journal: Trans. Amer. Math. Soc. 326 (1991), 345-369
MSC: Primary 20E06; Secondary 20F05, 20F06
DOI: https://doi.org/10.1090/S0002-9947-1991-1002920-5
MathSciNet review: 1002920
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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.


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Article copyright: © Copyright 1991 American Mathematical Society