Essential Closed Surfaces in Bounded Manifolds
by
Alan Reid
University of Texas
Coauthors: D. Cooper, D.D. Long

Let M be a finite volume hyperbolic 3-manifold whose boundary consists of a finite number (non-zero) of incompressible tori. An orientable closed surface S immersed in M is called essential if it is incompressible and not boundary parallel. A question dating back to Waldhausen and discussed in various contexts by Thurston is the problem of the extent to which irreducible 3-manifolds with infinite fundamental group must contain surface groups. Here we show:

Theorem: With M as above. Then \pi₁(M) has a finite index subgroup which maps onto a free group of rank 2.

In fact we show the following:

Theorem: M as above. Then M contains an immersed essential closed surface which lifts to an embedded non-separating surface in a finite cover.

It is a consequence of the proofs of these theorems that "non-peripheral homology" can be made arbitrarily large by passing to finite covers of M.

This is joint work with D. Cooper and D. D. Long.

Date received: February 14, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caab-86.