2005/07/14 by Marc Lackenby, Lackenby, Marc
Mathematics · #57M25 #57N10 #FOS: Mathematics #Geometric Topology (math.GT) #math.GT #msc:57M25 #msc:57N10
paper · pdf · doi:10.48550/arxiv.math/0507283
14 pages, 1 figure
arxiv created 2005/07/14 · arxiv updated 2009/12/01
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 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.