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?


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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.

  2. The number of boundary slopes (slopes whose multiples are boundaries of incompressible boundary incompressible surfaces) is finite. (Hatcher)

  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.


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