  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.
