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Transactions of the American Mathematical Society
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
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\vert 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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DOI: http://dx.doi.org/10.1090/S0002-9947-1991-1002920-5
PII: S 0002-9947(1991)1002920-5
Article copyright: © Copyright 1991 American Mathematical Society