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:


1. A compact 3-manifold with non-empty boundary is hyperbolizable if and only if it is irreducible and atoroidal.
2. Any compact, atoroidal, pared 3-manifold is diffeomorphic to a geometrically finite one.
3. Any compact hyperbolic 3-manifold is homotopy equivalent to a geometrically finite one.



Hyperbolic 3-manifolds with no geometrically finite structure
[Edited several times] As the comments say, the answer to the first and hence to the second question is "no". Suppose that $M$ is the compact manifold and $N$ is its interior. Let $\rho$ be the given hyperbolic structure on $N$. If $M$ is without boundary then the volume of $\rho$ is finite and we are done. 


Suppose instead that $M$ has boundary. Since $N$ is hyperbolic, via $\rho$, deduce $M$ is atoroidal (which includes aspherical). Thus $M$ is Haken. Place all tori in the boundary of $M$ into the paring locus $P$. By Thurston's hyperbolization theorem, the interior $N$ admits a hyperbolic metric, $\rho\_0$, which is geometrically finite. (The convex core has finite volume and contains all torus boundary components.) See Theorem 1.43 in Kapovich's book. 


[A brief note - your hypotheses can be weakened. You assumed (a) $N$ is the interior of a compact manifold and (b) N is hyperbolizable. This can be replaced by (a') $\pi\_1(N)$ is finitely generated and the same (b). This is called the "tameness theorem", due to Agol and also Calegari-Gabai.]


In the comments below (above?) Igor asks why an atoroidal manifold with torus boundary, and admitting an essential annulus, is Seifert fibered. This can be found as Lemma 1.16 on page 25 of Hatcher's three-manifold [notes](http://www.math.cornell.edu/~hatcher/3M/3Mfds.pdf).


