I just heard that Jeremy Kahn has announced joint work with Vladimir Markovic showing that every hyperbolic 3-manifold contains an immersed, -injective surface. This is equivalent to showing that the fundamental group of every hyperbolic 3-manifold contains a subgroup isomorphic to the fundamental group of a surface. The abstract for Jeremy’s talk is on the conference web page. This is a long standing open problem that Henry Wilton wrote about on this blog back in February. I’m not at the conference, so I didn’t see the talk, but if any of you readers were ther and have anything to add, feel free to write in in the comments.
August 5, 2009
Immersed surfaces with injective fundamental group.