Summary: Some surface subgroups survive surgery.
D. Cooper and D.D. Long \Lambda
March 23, 1998
1 Introduction.
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 [5]; 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.

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

Collections: Mathematics