Essential Immersed Surfaces in Closed Hyperbolic 3-Manifolds

Given any closed hyperbolic 3-manifold M and $\epsilon > 0$,
we find a closed hyperbolic surface S and a map $f: S \to M$
such that $f$ lifts to a $1+\epsilon$-quasi-isometry
from the universal cover of $S$ to the universal cover of $M$.
It follows that, for $\epsilon$ small, the map $f$ induces an injection on the fundamental group of $S$;
thus the fundamental group of every closed hyperbolic 3-manifold has a surface subgroup.
This is joint work with Vladimir Markovic.

I will explain why the mixing of the frame flow on $M$
implies the existence of a highly symmetric collection of pairs of pants,
which can then be assembled to form the desired surfaces $S$.