  A group is known as `large' if some finite index subgroup admits a surjective
homomorphism onto a non-abelian free group. The main theorem of the paper is as
follows. Let G be a finitely generated, large group and let g_1,...,g_r be a
collection of elements of G. Then G/<<g_1^n,...,g_r^n>> is also large, for
infinitely many integers n. Furthermore, when G is free, this holds for all but
finitely many n. These results have the following application to Dehn surgery.
Let M be a compact orientable 3-manifold with boundary a torus. Suppose that
the 3-manifold obtained by Dehn filling some slope on the boundary has large
fundamental group. Then this is true for infinitely many filling slopes.
