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 is a knot in a compact orientable 3-manifold such that the complement of admits a complete finite-volume hyperbolic structure, then the orbifold obtained by assigning a singularity of order along has large fundamental group for infinitely many positive integers . We also obtain information about this set of values of . When 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 is hyperbolic to the assumption that 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

Email:
lackenby@maths.ox.ac.uk

DOI:
http://dx.doi.org/10.1090/S0002-9939-07-09050-8

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.