Summary: Some surface subgroups survive surgery.
D. Cooper and D.D. Long \Lambda
March 23, 1998
A central unresolved question in the theory of closed hyperbolic 3manifolds is whether they are
covered by manifolds which contain closed embedded incompressible surfaces. An affirmative res
olution of this conjecture would imply in particular that all closed hyperbolic 3manifolds contain
the fundamental group of a closed surface of genus at least two. Even the simplest case of this
conjecture, namely that of the manifolds obtained by surgery on a hyperbolic manifold with a single
cusp has remained open for many years. In this article we prove the following theorem:
Theorem 1.1 Suppose that M is a hyperbolic 3manifold with a single torus boundary component.
Then all but finitely many surgeries on M contain the fundamental group of a closed orientable
surface of genus at least two.
Our proof rests upon:
Theorem 1.2 Suppose that S is an incompressible, @incompressible quasifuchsian surface with
boundary slope ff.
Then there is a K so that if fl is any simple curve on @M with \Delta(ff; fl) ? K, the Dehn filled
manifold M (fl) contains the fundamental group of a closed surface of genus at least two.
This result is similar in spirit to the main theorem of ; that result applied only to surfaces of slope
zero, however in that context we were able to give an explicit (and fairly small) value for K.