KahnMarkovic show that every hyperbolic 3fold contains an immersed $\pi_1$ injective surface. Are there any known examples of hyperbolic 3folds 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:
Take a hyperbolic filling on a nonboundary 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. 

