Does there exist a compact hyperbolic 3-manifold $M$ that is not diffeomorphic to a geometrically finite hyperbolic manifold? If yes, can such $M$ have incompressible boundary?
I think the answer should be yes to both questions but I cannot find this in the literature.
Remarks: as usual, a compact hyperbolic manifold is a compact manifold whose interior carries a complete hyperbolic structure. The structure is geometrically finite if it is obtained as the quotient of the hyperbolic 3-space by a geometrically finite group. Thurston's hyperbolization theorem implies:
A compact 3-manifold with non-empty boundary is hyperbolizable if and only if it is irreducible and atoroidal.
Any compact, atoroidal, pared 3-manifold is diffeomorphic to a geometrically finite one.
Any compact hyperbolic 3-manifold is homotopy equivalent to a geometrically finite one.