Essential closed surfaces in bounded 3-manifolds

Abstract

A question dating back to Waldhausen [10] and discussed in various contexts by Thurston (see [9]) is the problem of the extent to which irreducible 3-manifolds with infinite fundamental group must contain surface groups. To state our results precisely, it is convenient to make the definition that a map i: S 9<M of a closed, orientable connected surface S is essential if it is injective at the level of fundamental groups and the group i*rr1 (S) cannot be conjugated into a subgroup 7rr(coM) of -rr(M), where &oM is a component of OM. This latter condition is equivalent to the statement that the image of the surface S cannot be freely homotoped into OM. One of the main results of this paper is the following: