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Essential closed surfaces in bounded 3manifolds. D. Cooper, D.D. Long and A. W. Reid \Lambda

Summary: Essential closed surfaces in bounded 3­manifolds.
D. Cooper, D.D. Long and A. W. Reid \Lambda
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 3\Gammamanifolds 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 i \Lambda ß 1 (S) cannot be conjugated into a subgroup ß 1 (@ 0 M)
of ß 1 (M ), where @ 0 M 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 3­manifold with nonempty incompressible boundary. Suppose
that the interior of M has a complete hyperbolic structure.
Then either M is covered by a product F \Theta I where F is a closed orientable surface or M contains
an essential surface S of genus at least 2:
Furthermore S can lifted to an embedded nonseparating surface in a finite cover of M:
In particular this gives a complete resolution to Waldhausen and Thurston's questions in the
context of manifolds with non­empty incompressible boundary:
Corollary 1.2 Let M be a compact irreducible manifold with nonempty incompressible boundary.
Then either M is covered by a product F \Theta I or M contains an essential closed surface.


Source: Akhmedov, Azer - Department of Mathematics, University of California at Santa Barbara


Collections: Mathematics