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

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.