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Some 3-manifolds and 3-orbifolds with large fundamental group

Author: Marc Lackenby
Journal: Proc. Amer. Math. Soc. 135 (2007), 3393-3402
MSC (2000): Primary 57N10, 57M25
Published electronically: June 22, 2007
MathSciNet review: 2322772
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Abstract: We provide two new proofs of a theorem of Cooper, Long and Reid which asserts that, apart from an explicit finite list of exceptional manifolds, any compact orientable irreducible 3-manifold with non-empty boundary has large fundamental group. The first proof is direct and topological; the second is group-theoretic. These techniques are then applied to prove a string of results about (possibly closed) 3-orbifolds, which culminate in the following theorem. If $ K$ is a knot in a compact orientable 3-manifold $ M$ such that the complement of $ K$ admits a complete finite-volume hyperbolic structure, then the orbifold obtained by assigning a singularity of order $ n$ along $ K$ has large fundamental group for infinitely many positive integers $ n$. We also obtain information about this set of values of $ n$. When $ M$ is the 3-sphere, this has implications for the cyclic branched covers over the knot. In this case, we may also weaken the hypothesis that the complement of $ K$ is hyperbolic to the assumption that $ K$ is non-trivial.

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Additional Information

Marc Lackenby
Affiliation: Mathematical Institute, University of Oxford, 24-29 St Giles, Oxford OX1 3LB, United Kingdom

Received by editor(s): May 12, 2006
Published electronically: June 22, 2007
Additional Notes: The author was supported by the EPSRC
Communicated by: Daniel Ruberman
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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