 
Summary: JOURNAL OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 10, Number 3, July 1997, Pages 553563
S 08940347(97)002361
ESSENTIAL CLOSED SURFACES IN BOUNDED 3MANIFOLDS
D. COOPER, D. D. LONG, AND A. W. REID
1. Introduction
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 3manifolds
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 M of a closed,
orientable connected surface S is essential if it is injective at the level of fundamental
groups and the group i1(S) cannot be conjugated into a subgroup 1(0M) of
1(M), where 0M is a component of M. This latter condition is equivalent to
the statement that the image of the surface S cannot be freely homotoped into M.
One of the main results of this paper is the following:
Theorem 1.1. Let M be a compact, connected 3manifold with nonempty incom
pressible boundary. Suppose that the interior of M has a complete hyperbolic struc
ture. Then either M is covered by a product F × I, where F is a closed orientable
surface, or M contains an essential surface S of genus at least 2.
