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Summary: RESEARCH BLOG 3/29/04
SURFACES IN 3-MANIFOLDS
At the AMS meeting in Tallahassee, Joe Masters spoke about joint
work with Xingru Zhang. They show that if a 1-cusped hyperbolic 3-
manifold has a totally geodesic immersed surface, then there is a closed
quasifuchsian surface. It is conjectured that this holds true in general,
and they hope to extend their result by using incompressible surfaces
with boundary provided by Culler and Shalen's theorem. Cooper and
Long (and independently Li) showed that there are closed surfaces, by
using a spinning argument generalizing work of Freedman and Freed-
man, but this construction gives geometrically finite surfaces with ac-
cidental parabolics. They are still able to use this to show that all but
finitely many Dehn fillings on a cusp of a hyperbolic manifold contain
closed 1-injective surfaces. In fact, their construction gives surfaces
with only double curves. One can generalize this slightly to show that
if for a given number V > 0, one considers hyperbolic 3-manifolds of
volume < V , then all but finitely many contain a 1-injective surface
with only double curves. This is in stark contrast to Haken manifolds,
of which there can be infinitely many of bounded volume. It would
be interesting to find hyperbolic 3-manifolds which do not contain any
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