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Transactions of the American Mathematical Society

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

 
 

 

Frattini subgroups of $ 3$-manifold groups


Authors: R. B. J. T. Allenby, J. Boler, B. Evans, L. E. Moser and C. Y. Tang
Journal: Trans. Amer. Math. Soc. 247 (1979), 275-300
MSC: Primary 57M05; Secondary 20F34
DOI: https://doi.org/10.1090/S0002-9947-1979-0517695-8
MathSciNet review: 517695
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Abstract: In this paper it is shown that if the Frattini subgroup of the fundamental group of a compact, orientable, irreducible, sufficiently large 3-manifold is nontrivial then the 3-manifold is a Seifert fibered space. We show further that the Frattini subgroup of the group of a Seifert fibered space is trivial or cyclic. As a corollary to our work we prove that every knot group has trivial Frattini subgroup.


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DOI: https://doi.org/10.1090/S0002-9947-1979-0517695-8
Keywords: Frattini subgroup, generalized free product with amalgamation, HNN extension, incompressible 2-manifold, sufficiently large 3-manifold, Seifert fibered space, irreducible 3-manifold
Article copyright: © Copyright 1979 American Mathematical Society

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