# Example of hyperbolic 3-fold with no embedded incompressible subsurfaces

Kahn-Markovic show that every hyperbolic 3-fold contains an immersed $\pi_1$ injective surface. Are there any known examples of hyperbolic 3-folds that do not contain a embedded $\pi_1$ injective surface?

-
Chris, these are called non-Haken manifolds (or orbifolds). There are many examples known. Here's some references: Ch. 4 of Thurston's notes ams.org/mathscinet-getitem?mr=2026837 (this has a literature survey, see citations therein) front.math.ucdavis.edu/0909.4625 –  Ian Agol Dec 14 '11 at 19:00
Thanks Ian. This is very helpful. –  Chris Judge Dec 15 '11 at 10:03

Infinitely many Dehn fillings on the figure eight knot complement $M_8$ have this property:
1. All but finitely many fillings on $M_8$ are hyperbolic, by Thurston's hyperbolic Dehn filling theorem.
3. There are no embedded closed incompressible non-peripheral surfaces in $M_8$ (originally checked by Thurston?)
Take a hyperbolic filling on a non-boundary slope. If this were Haken, then, by a cut and paste argument, any incompressible surface $F$ could be made to intersect $M_8$ in an incompressible boundary incompressible surface, contradicting the choice of slope.