Some 3manifolds and 3orbifolds with large fundamental group
Author:
Marc Lackenby
Journal:
Proc. Amer. Math. Soc. 135 (2007), 33933402
MSC (2000):
Primary 57N10, 57M25
Published electronically:
June 22, 2007
MathSciNet review:
2322772
Fulltext PDF Free Access
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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 3manifold with nonempty boundary has large fundamental group. The first proof is direct and topological; the second is grouptheoretic. These techniques are then applied to prove a string of results about (possibly closed) 3orbifolds, which culminate in the following theorem. If is a knot in a compact orientable 3manifold such that the complement of admits a complete finitevolume 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 3sphere, 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 nontrivial.
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Additional Information
Marc Lackenby
Affiliation:
Mathematical Institute, University of Oxford, 2429 St Giles, Oxford OX1 3LB, United Kingdom
Email:
lackenby@maths.ox.ac.uk
DOI:
http://dx.doi.org/10.1090/S0002993907090508
PII:
S 00029939(07)090508
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.
